2. Preliminaries

Klein surfaces constitute a generalization of Riemann surfaces that include bordered and nonorientable surfaces. They broaden the scope of Riemann surfaces by allowing transition functions that may include complex conjugation besides analytic functions and domains in the closed upper half-plane ${\mathbb{C}}^+$ . This makes up what is called a dianalytic structure [Reference Alling and Greenleaf1]. The topological genus g, the number k of boundary components, and the orientability are known as the topological type of a Klein surface, and the integer $p=\eta g+k-1$ as its algebraic genus, where $\eta=2$ if the surface is orientable and $\eta=1$ otherwise. By a nonorientable Riemann surface, we mean a nonorientable unbordered Klein surface.

A non-Euclidean crystallographic (NEC) group ${\Lambda}$ is a discrete subgroup of the group of isometries of the hyperbolic plane ${\mathcal{H}}$ for which ${\mathcal{H}}/{\Lambda}$ is compact. It is a Fuchsian group if it contains only orientation preserving isometries; otherwise, it is said to be a proper NEC group. An NEC group with no orientation preserving elements of finite order is called surface NEC group.

Every compact Klein surface with algebraic genus $p>1$ can be represented by the orbit space ${\mathcal{H}}/{\Gamma}$ for some surface NEC group ${\Gamma}$ , that is, surface NEC groups uniformize compact Klein surfaces. Furthermore, every group G of automorphisms of ${\mathcal{H}}/{\Gamma}$ is isomorphic to the factor group ${\Lambda}/{\Gamma}$ for some NEC group ${\Lambda}$ containing the surface NEC group ${\Gamma}$ as a normal subgroup, that is, there is an epimorphism $\theta\,:\,{\Lambda}\to G$ for which $\ker\theta={\Gamma}$ (we say that $\theta$ is a surface-kernel epimorphism).

It is well-known that every group of finite order acts on some compact Klein surface of algebraic genus greater than 1 [Reference Bujalance4]. A group may act on Klein surfaces of different genera. The minimum genus problem consists in finding the least genus on which a group acts. When dealing with nonorientable Riemann surfaces, such minimum (topological) genus is called symmetric cross-cap number of the group, and is denoted by $\widetilde{\sigma}(G)$ . Conversely, several groups may act on some Klein surface of a given algebraic genus. When the algebraic genus is greater than 1, there are only finitely many such groups. Computing the largest group order in a family of groups which act on a given genus is what we call the maximum order problem for that family.

Non-isomorphic NEC groups differ from one another in the signature, which is of the form

(2.1) \begin{equation}\left(g;\,\pm;\,[m_1,\ldots,m_r];\,\left\{(n_{i1},\ldots,n_{is_i}), \,i=1, \ldots, k\right\}\right).\end{equation}

The signature of a Fuchsian group is usually denoted by $(g;\,m_1,\ldots,m_r)$ . For a surface NEC group, it is of the form $(g;\,\pm;$ $\,[{-}];\,\{({-}), \overset{k}{\ldots}, ({-})\})$ . The signature of an NEC group ${\Lambda}$ determines both its algebraic structure and the topological structure of the orbit space ${\mathcal{H}}/{\Lambda}$ .

The integers $m_i\geqslant 2$ are called proper periods, $n_{ij}\geqslant 2$ are the link periods, $(n_{i1},\ldots,$ $n_{is_i})$ are the period cycles and g is the orbit genus. The orbit space ${\mathcal{H}}/{\Lambda}$ has topological genus g, k boundary components and is orientable if the sign of the signature is ‘ $+$ ’ and nonorientable otherwise. The covering map ${\mathcal{H}}\to{\mathcal{H}}/{\Lambda}$ ramifies over r interior points with ramification indices $m_i$ and, on each boundary component, over $s_i$ points with ramification indices $n_{ij}$ . The number $\eta g+k-1$ is the algebraic genus of ${\mathcal{H}}/{\Lambda}$ , where $\eta=2$ if the sign of the signature is ‘ $+$ ’ and $\eta=1$ otherwise. An arbitrary set of such numbers and symbols define the signature of an NEC group if and only if

(2.2) \begin{equation}\eta g+k-2 + \sum_{i=1}^{r} \left( 1-\frac{1}{m_i} \right) + \frac{1}{2} \sum_{i=1}^{k} \sum_{j=1}^{s_i} \left( 1-\frac{1}{n_{ij}} \right) > 0.\end{equation}

The expression in the left side is denoted by $\mu({\Lambda})$ . The hyperbolic area of any fundamental region of ${\mathcal{H}}/{\Lambda}$ is $2\pi\mu({\Lambda})$ . Also, if ${\Lambda}'$ is a subgroup of ${\Lambda}$ of finite index, then ${\Lambda}'$ is an NEC group and

(2.3) \begin{equation}[{\Lambda}\,:\,{\Lambda}'] = \mu({\Lambda}')/\mu({\Lambda}),\end{equation}

which is known as the Riemann–Hurwitz formula.

The signature provides a presentation of ${\Lambda}$ with the following generators and relations depending on the sign of the signature:

\begin{align*}& x_1,\ldots,x_r \qquad\qquad\qquad\qquad\qquad\qquad \text{(hyperbolic rotations)}, \\& c_{10},\ldots,c_{1s_1},\ldots,c_{k0},\ldots,c_{ks_k} \qquad\qquad\, \text{(hyperbolic reflections)}, \\& e_1,\ldots,e_k \qquad\qquad\qquad\qquad\qquad\qquad \text{(connecting generators)}, \\& a_1, b_1,\ldots,a_g, b_g \quad\text{if the sign is `+'} \qquad\text{(hyperbolic translations)}, \\& d_1,\ldots,d_g \qquad\quad\,\,\,\,\text{if the sign is `$-$'} \qquad \text{(hyperbolic glide reflections)},\end{align*}

\begin{equation*}x_i^{m_i}=1, \qquad c_{ij}^2=1, \qquad (c_{ij-1}c_{ij})^{n_{ij}}=1, \qquad e_i^{-1}c_{i0}e_ic_{is_i}=1,\end{equation*}

\begin{equation*}x_1\cdots x_re_1\cdots e_ka_1b_1a_1^{-1}b_1^{-1}\cdots a_gb_ga_g^{-1}b_g^{-1}=1 \qquad \text{if the sign is `$+$' and}\end{equation*}

\begin{equation*}x_1\cdots x_re_1\cdots e_kd_1^2\cdots d_g^2=1 \qquad \text{if the sign is `$-$'.}\end{equation*}

The last one is called the long relation. (An abstract group with such a presentation is an NEC group with signature as above if and only if (2.2) is fulfilled.)

For further purposes, the following should be considered. For proper periods, we assume factorizations $m_i=p_1^{\mu_{i}(p_1)}\cdots p_s^{\mu_{i}(p_s)}$ with prime numbers $p_1<\cdots<p_s$ and integers $\mu_{i}(p_j)\geqslant 0$ such that $\mu_1(p_j)+\cdots+\mu_r(p_j)>0$ . For each prime $p_j$ , we rearrange the integers $\mu_1(p_j), \ldots, \mu_r(p_j)$ to obtain increasing integers ${\widehat\mu}_{1}(p_j)\leqslant{\widehat\mu}_{2}(p_j)\leqslant\cdots\leqslant{\widehat\mu}_{r}(p_j)$ and define ${\widehat m}_i=p_1^{{\widehat\mu}_{i}(p_1)}\cdots p_s^{{\widehat\mu}_{i}(p_s)}$ . Then, ${\widehat m}_i|{\widehat m}_{i+1}$ and there is an integer ${\widehat r}$ such that ${\widehat m}_i=1$ for $i=1,\ldots,r-{\widehat r}$ and ${\widehat m}_i>1$ for the ${\widehat r}$ integers $i=r-{\widehat r}+1,\ldots,r$ . Moreover (see [Reference Rodríguez13, Section 2]),

(2.4) \begin{equation}\sum_{i=1}^{r}\left(1-\frac{1}{{\widehat m}_i}\right) \;\leqslant\; \sum_{i=1}^{r}\left(1-\frac{1}{m_i}\right).\end{equation}

Henceforth, we will deal with Abelian groups. The torsion subgroup of an Abelian group A is denoted by $\mathcal{T}(A)$ . When A is a finitely generated Abelian group, its invariant factor decomposition is $A\approx{\mathbb{Z}}^n\oplus{\mathbb{Z}}_{{v}_1}\oplus\cdots\oplus{\mathbb{Z}}_{{v}_t}$ for integers $n\geqslant 0$ , the torsion-free rank of A, and ${v}_i>1$ , called invariant factors of A, with ${v}_i$ dividing ${v}_{i+1}$ , and primary decomposition $A\approx {\mathbb{Z}}^n\oplus A_{q_1}\oplus\cdots\oplus A_{q_\lambda}$ , where $q_1<\cdots<q_\lambda$ are the prime numbers dividing the order of A and $A_q=\{x\in A | q^mx=0 \text{ for some } m\geqslant 0\}$ is the q-primary component of A —the q-Sylow subgroup $Syl_q(A)$ . We also assume ${v}_i=q_1^{\alpha_i(q_1)}\cdots q_\lambda^{\alpha_i(q_\lambda)}$ for $i=1,\ldots,t$ , so $0\leqslant\alpha_1(q)\leqslant\cdots\leqslant\alpha_t(q)$ and $A_q\approx {\mathbb{Z}}_{q^{\alpha_1(q)}} \oplus\cdots\oplus {\mathbb{Z}}_{q^{\alpha_t(q)}}$ . The integers $q_j^{\alpha_i(q_j)}$ are the elementary divisors of A.

Below it will be helpful to express a finite Abelian group as follows:

\begin{equation*}A\approx {\mathbb{Z}}_2^{\,n}\oplus{\mathbb{Z}}_{{v}_1}\oplus\cdots\oplus{\mathbb{Z}}_{{v}_t},\end{equation*}

(for readability, ${\mathbb{Z}}_v\oplus\overset{n}{\cdots}\oplus{\mathbb{Z}}_v$ will be denoted by ${\mathbb{Z}}_v^{\;n}$ ) where $v_i>2$ and $v_i$ divides $v_{i+1}$ , so that $v_1,\ldots,v_{t-m}$ are odd and the m integers $v_{t-m+1},\ldots,v_t$ are multiple of 4 for some integer $m\leqslant t$ —note that, though unique, this expression may not coincide with the invariant factor decomposition of A.

3. Abelianization of NEC groups

In this section, we lay out Breuer conditions for the existence of epimorphisms between Abelian groups and the abelianization of NEC groups.

We are mainly concerned with conditions for the existence of epimorphisms ${\theta}\,:\,{\Lambda}\to A$ from an NEC group onto a finite Abelian group. In this context, the Abelianization ${\Lambda}_{ab}$ of ${\Lambda}$ provides significant information. By the universal property of the quotient group, ${\theta}$ factors (uniquely) through the canonical homomorphism $\pi\,:\,{\Lambda}\to{\Lambda}_{ab}$ . In other words, there is a (unique) epimorphism $\overline{{\theta}}\,:\,{\Lambda}_{ab}\to A$ such that ${\theta}=\overline{{\theta}}\circ\pi$ . Conversely, given an epimorphism $\overline{{\theta}}\,:\,{\Lambda}_{ab}\to A$ , the homomorphism ${\theta}=\overline{{\theta}}\circ\pi\,:\,{\Lambda}\to A$ is onto.

Therefore, it is worth considering epimorphisms between Abelian groups. Breuer stated conditions for the existence of such epimorphisms as a set of inequations on the free-rank and the number of cyclic factors in the primary decomposition of the Abelian groups [Reference Breuer2, lemmas A.1 and A.2]:

Lemma 3.1. Let q be a prime number and $R, N_1,$ $\ldots,$ $N_s, T, n_1,\ldots,$ $n_s$ be non-negative integers. There is an epimorphism

\begin{equation*}{\mathbb{Z}}^R\oplus{\mathbb{Z}}_{q^1}^{\,N_1}\oplus\cdots\oplus{\mathbb{Z}}_{q^s}^{\,N_s} \quad\to\quad {\mathbb{Z}}^T\oplus{\mathbb{Z}}_{q^1}^{\,n_1}\oplus\cdots\oplus{\mathbb{Z}}_{q^s}^{\,n_s}\end{equation*}

if and only if

(3.1) \begin{align}R \;\geqslant\; T \quad \text{and} \quad R+N_j+N_{j+1}+\cdots+N_s \;\geqslant\; &T+n_j+n_{j+1}+\cdots+n_s \notag\\ & \text{for} \quad j=1,\ldots,s.\end{align}

For arbitrary finite Abelian groups A and B, there is an epimorphism ${\mathbb{Z}}^R\oplus A \to {\mathbb{Z}}^T\oplus B$ if and only if there is an epimorphism ${\mathbb{Z}}^R\oplus A_q \to {\mathbb{Z}}^T\oplus B_q$ for each prime q dividing the order of B.

The requirements in (3.1) can be rewritten as follows.

Lemma 3.2. Let q be a prime number and $R, \alpha_1, \ldots,$ $\alpha_t, \beta_1,\ldots,$ $\beta_r$ be non-negative integers with $\alpha_i\leqslant\alpha_{i+1}$ and $\beta_i\leqslant\beta_{i+1}$ . There is an epimorphism

\begin{equation*}{\mathbb{Z}}^R\oplus{\mathbb{Z}}_{q^{\beta_1}}\oplus\cdots\oplus{\mathbb{Z}}_{q^{\beta_r}} \quad\to\quad {\mathbb{Z}}^T\oplus{\mathbb{Z}}_{q^{\alpha_1}}\oplus\cdots\oplus{\mathbb{Z}}_{q^{\alpha_t}}\end{equation*}

if and only if the following conditions hold: $R\geqslant T$ and if $R<T+t$ , then $q^{\alpha_i}$ divides, at least, $T+t-R-i+1$ elementary divisors $q^{\beta_j}$ for $i=1,\ldots,T+t-R$ .

In order to study these conditions for epimorphisms ${\Lambda}_{ab}\to A$ , we need to know the structure of ${\Lambda}_{ab}$ in terms of the signature of ${\Lambda}$ . Here a distinction is made between Fuchsian and proper NEC groups. For a Fuchsian group ${\Lambda}$ , we find its Abelianization in [Reference Breuer2, Lemma A.3].

Lemma 3.3. The Abelianization of a Fuchsian group ${\Lambda}$ with signature $(g;\,m_1,$ $\ldots,$ $m_r)$ is isomorphic to ${\mathbb{Z}}^{2g}$ if $r=0$ or 1 and

\begin{equation*}{\Lambda}_{ab} \,\approx\, {\mathbb{Z}}^{2g} \oplus {\mathbb{Z}}_{{\widehat m}_{r-{\widehat r}+1}} \oplus \cdots \oplus {\mathbb{Z}}_{{\widehat m}_{r-1}}\end{equation*}

otherwise.

Now, we compute the Abelianization ${\Lambda}_{ab}$ of a proper NEC group ${\Lambda}$ . When the signature has some period cycle, ${\Lambda}_{ab}$ is obtained by some considerations on the canonical presentation of ${\Lambda}$ . Otherwise, it has no period cycle and we compute the Smith normal form ([Reference Smith16][Reference Magnus, Karrass and Solitar11, Section 3.3][Reference Newman12, Chapter 2]) of the relation matrix of the Abelianized canonical presentation of ${\Lambda}$ .

Lemma 3.4. The Abelianization of a proper NEC group ${\Lambda}$ with signature $(g;\,\pm;$ $[m_1,\ldots,m_r];\,\{(n_{i1},\ldots,n_{is_i}), \,i=1, \ldots, k\})$ is ${\Lambda}_{ab} \,\approx\, {\mathbb{Z}}^{\eta g+k-1} \oplus \mathcal{T}({\Lambda}_{ab})$ , where

(3.2) \begin{equation}\mathcal{T}({\Lambda}_{ab}) \,\approx\; \begin{cases}{\mathbb{Z}}_2 & \text{if $k=r=0$,} \\[1mm]{\mathbb{Z}}_{{\widehat m}_{r-{\widehat r}+1}} \oplus \cdots \oplus {\mathbb{Z}}_{{\widehat m}_{r-1}} \oplus {\mathbb{Z}}_{2{\widehat m}_r} & \text{if $k=0$, $r>0$,} \\[1mm]{\mathbb{Z}}_2^{\;S} \oplus {\mathbb{Z}}_{{\widehat m}_{r-{\widehat r}+1}} \oplus \cdots \oplus {\mathbb{Z}}_{{\widehat m}_r} & \text{otherwise}\end{cases}\end{equation}

is the torsion subgroup of ${\Lambda}_{ab}$ , $\eta$ equals 2 if the sign of the signature is ‘ $+$ ’ and 1 otherwise, and

\begin{equation*}S=\#\{\text{period cycles with no even link periods}\}\,+\,\#\{\text{even link periods}\}.\end{equation*}

Proof. When $k>0$ , we remove one generator $e_i$ by the long relation in the Abelianized presentation, and the other relations only contain generators of finite order. The remaining canonical generators $e_i$ , $a_i$ , $b_i$ and $d_i$ provide the factor ${\mathbb{Z}}^{\eta g+k-1}$ , while the elliptic generators and their relations turn into ${\mathbb{Z}}_{m_{1}} \oplus \cdots \oplus {\mathbb{Z}}_{m_r} \approx {\mathbb{Z}}_{{\widehat m}_{r-{\widehat r}+1}} \oplus \cdots \oplus {\mathbb{Z}}_{{\widehat m}_r}$ .

The factor ${\mathbb{Z}}_2^{\;S}$ originates from the generators $c_{ij}$ . There are $k+s_1+\cdots+s_k$ of such generators; we remove k of them (each relation $e_i^{-1}c_{i0}e_ic_{is_i}=1$ lets us remove $c_{i0}$ or $c_{is_i}$ when Abelianized) and also those generators $c_{ij}$ for which $n_{ij}$ is odd (when Abelianized, the relation $(c_{ij-1}c_{ij})^{n_{ij}}=1$ becomes either $c_{ij-1}c_{ij}=1$ or trivial for odd and even values of $n_{ij}$ , respectively).

If $k=0$ (hence the sign of the signature is ‘–’ since ${\Lambda}$ is proper), a presentation of ${\Lambda}_{ab}$ is

\begin{equation*}\langle x_1,\ldots,x_r,d_1,\ldots,d_g \;|\; x_i^{m_i}, \;x_1\cdots x_rd_1^2\cdots d_g^2,[x_i,x_j],[x_i,d_j],[d_i,d_j]\rangle.\end{equation*}

The relation matrix of this presentation is

\begin{equation*}\textbf{R} = \left(\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}m_1 & & & \\[3pt]& & m_2 & & & & 0 \\[3pt] & & & \ddots & & \\[1pt] & 0 & & & \\[3pt] & & & & m_r & \\[3pt] 1 & & \cdots & 1 & 1 & 2 & & \overset{g}{\cdots} & 2\end{array}\right).\end{equation*}

The entries $\epsilon_i$ in the main diagonal of the Smith normal form of $\textbf{R}$ can be computed as follows: $\epsilon_1=\rho_1$ , $\epsilon_2=\rho_2/\rho_1$ , $\epsilon_3=\rho_3/\rho_2$ , etc., where $\rho_i$ is the greatest common divisor of the minor determinants of order i of $\textbf{R}$ .

Clearly, $\rho_1=1$ . Non-null $2\times 2$ minors of $\textbf{R}$ are $m_i, 2m_i$ or $m_{i_1}m_{i_2}$ and thus $\rho_2=\gcd(m_1,\ldots,m_r)={\widehat m}_1$ . Likewise, non-null $3\times 3$ minors take values $m_{i_1}m_{i_2}$ , $2m_{i_1}m_{i_2}$ or $m_{i_1}m_{i_2}m_{i_3}$ so that $\rho_3=\gcd(m_1m_2,\ldots,m_1m_r, m_2m_3,$ $\ldots, m_{r-1}m_r)={\widehat m}_1{\widehat m}_2$ . In general,

\begin{align*}\rho_k &= \gcd\{m_{i_1}\cdots m_{i_{k-1}}\}_{1\leqslant i_1<i_2<\cdots<i_{k-1}\leqslant r} = {\widehat m}_1\cdots{\widehat m}_{k-1}, \\\rho_{r+1} &= 2m_1\cdots m_r = 2{\widehat m}_1\cdots{\widehat m}_r\end{align*}

for $k=2,\ldots,r$ and thus $\epsilon_1 = 1, \;\epsilon_2 = {\widehat m}_1, \;\epsilon_3 = {\widehat m}_2, \; \ldots, \;\epsilon_{r} = {\widehat m}_{r-1}, \;\epsilon_{r+1} = 2{\widehat m}_r.$ Therefore, ${\Lambda}_{ab} \,\approx\, {\mathbb{Z}}^{g-1} \oplus {\mathbb{Z}}_{{\widehat m}_{r-{\widehat r}+1}} \oplus \cdots \oplus {\mathbb{Z}}_{{\widehat m}_{r-1}} \oplus {\mathbb{Z}}_{2{\widehat m}_r}$ when $k=0$ .

4. Nonorientable unbordered surface-kernel epimorphisms

In this section, conditions for the existence of nonorientable unbordered surface-kernel epimorphisms onto an Abelian group are established in terms of its algebraic structure.

We say that a homomorphism ${\theta}$ of a proper NEC group ${\Lambda}$ into a finite group G is nonorientable unbordered surface-kernel if $\ker{\theta}$ is a surface group with signature having sign ‘–’ and no period cycles. The condition on the sign of the signature means that ${\theta}({\Lambda}^+)=G$ , as stated by Singerman [Reference Singerman15, Theorem 1]. As a consequence, the following result establishes the type of signatures we will deal with.

Remark 4.2. In order to prove Theorem 4.3, we need to make some considerations on the Smith normal form of (4.1), which is obtained as follows. There is some orientation-reversing element in the kernel of a nonorientable unbordered surface-kernel epimorphism ${\theta}\,:\,{\Lambda}\to A$ onto a finite Abelian group A. Since ${\theta}(c_{i0})={\theta}(c_{is_i})$ , we can reorder the product of canonical generators in the expression of such orientation-reversing element to obtain another orientation-reversing element h with the same image and an expression like the following:

\begin{equation*}h=\begin{cases}h'\cdot c_{10}^{\kappa_1}\cdots c_{1,s_1-1}^{\kappa_{s_1}}\cdot c_{2,0}^{\kappa_{s_1+1}} \cdots c_{k,s_k-1}^{\kappa_S} & \\[3mm]h'\cdot d_1^{\zeta_1}\cdots d_g^{\zeta_g}\cdot c_{10}^{\kappa_1}\cdots c_{1,s_1-1}^{\kappa_{s_1}}\cdot c_{2,0}^{\kappa_{s_1+1}} \cdots c_{k,s_k-1}^{\kappa_S} &\end{cases}\end{equation*}

for signature sign ‘ $+$ ’ or ‘–’, respectively, without any of the k canonical reflections $c_{is_i}$ , where h’ is a product of powers of orientation-preserving canonical generators, $\kappa_1+\cdots+\kappa_S$ or $\zeta_1+\cdots+\zeta_g+\kappa_1+\cdots+\kappa_S$ , depending on the signature sign, is odd, and $\kappa_i=0$ or 1 since $c_{ij}^2=1$ .

We can factor ${\theta}$ through $N=\left\langle h \right\rangle^{\Lambda}$ , the normal subgroup generated by h. By the universal property of the quotient group, there exists a unique homomorphism $\phi\,:\,{\Lambda}/N\to A$ such that $\phi\circ\pi={\theta}$ , where $\pi\,:\,{\Lambda}\to{\Lambda}/N$ is the canonical projection. Since ${\theta}$ and $\pi$ are onto, $\phi$ is onto as well. Now, we factor $\phi$ through $({\Lambda}/N)_{ab}$ to obtain another epimorphism $\overline{\phi}\,:\,({\Lambda}/N)_{ab}\to A$ . We will extract some information from this epimorphism between Abelian groups by applying lemmas 3.1 and 3.2. It is not mandatory to compute $({\Lambda}/N)_{ab}$ ; the following considerations will be enough for our purposes.

By Lemma 4.1, we will focus on proper NEC groups with signatures of type $(g;\,\pm;\,[m_1,$ $\ldots,$ $m_r];\,$ $\{({-})^{{\varepsilon}},$ $(2,\overset{s_{{\varepsilon}+1}}{\ldots},2), \ldots, (2,\overset{s_k}{\ldots},2)\})$ . In that case, a presentation of $({\Lambda}/N)_{ab}$ is

\begin{align*}\langle x_1,\ldots,x_r,a_1,b_1,\ldots,&a_g,b_g,e_1,\ldots,e_k,c_{10},\ldots,c_{1,s_1-1},c_{20},\ldots,c_{k,s_k-1} \;|\; \\& x_i^{m_i}, \;x_1\cdots x_re_1\cdots e_k,c_{ij}^2,h,[\cdot,\cdot]\rangle\end{align*}

(by abuse of notation, we use the same symbols for the generators) if the signature sign is ‘ $+$ ’ and

\begin{align*}\langle x_1,\ldots,x_r,d_1,\ldots,& \,d_g,e_1,\ldots,e_k,c_{10},\ldots,c_{1,s_1-1},c_{20},\ldots,c_{k,s_k-1} \;|\; \\& x_i^{m_i}, \;x_1\cdots x_re_1\cdots e_kd_1^2\cdots d_g^2,c_{ij}^2,h,[\cdot,\cdot]\rangle\end{align*}

if the signature sign is ‘–’, where $[\cdot,\cdot]$ denotes all commutation relations of pairs of generators. We have removed the generators $c_{is_i}$ (by means of the Abelianized relations $e_i^{-1}c_{i0}e_ic_{is_i}=1$ ) and the relations $(c_{ij-1}c_{ij})^2=1$ (they become trivial when Abelianized). The relation matrix of this presentation is

(4.1) \begin{equation}\left(\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}m_1 & & & & & 0 \\[3pt] & \ddots & & \\[3pt] & & m_r & \\[3pt] 1 & \cdots & 1 & 1 & \displaystyle\overset{k}{\cdots} & 1 & 2 & \overset{g}{\cdots} & 2 & 0 & \overset{S}{\cdots} & 0 \\[3pt] & \cdots & & & \cdots & & \zeta_1 & \cdots & \zeta_g & \kappa_1 & \cdots & \kappa_S \\[3pt] & & & &&&&& & 2 \\[3pt] &&&&0 &&&&&& \ddots &\\[3pt] &&&&&&&&&&& 2\end{array}\right) \end{equation}

if the signature sign is ‘–’; if the signature sign is ‘ $+$ ’, remove the g columns containing the submatrix $\left(\begin{smallmatrix}2 & \overset{g}{\cdots} & 2 \\\zeta_1 & \cdots & \zeta_g\end{smallmatrix}\right)$ (to lessen clutter, the 2g columns corresponding to the canonical generators $a_i$ and $b_i$ are omitted, since they consist of null entries, and also entries corresponding to orientation-preserving generators in the $(r+2)$ nd row are not denoted).

Proof. Let ${\theta}\,:\,{\Lambda}\to A$ be a nonorientable unbordered surface-kernel epimorphism.

1. (i) The order of ${\theta}(x_i)$ is $m_i$ and the order of every element of A divides the exponent of A ( $\exp A=2$ if $t=0$ , $v_t$ if $n=0$ and $lcm(2,v_t)$ otherwise).

2. (ii) This condition follows from Lemma 3.2 applied to the epimorphism ${\Lambda}_{ab}\to A$ (we note that $v_i$ is either odd or multiple of 4, so that we do not have to consider the factors ${\mathbb{Z}}_2$ in (3.2)).

3. (iii) Let $k=0$ . If $g=1$ , then $\zeta_1$ is odd in (4.1) and it follows that $({\Lambda}/N)_{ab}$ has null free-rank and $Syl_2(({\Lambda}/N)_{ab})\approx{\mathbb{Z}}_{2^{{\widehat\mu}_1(2)}} \oplus \cdots \oplus {\mathbb{Z}}_{2^{{\widehat\mu}_{r-1}(2)}}$ . Now, assume that $g>1$ and let $2^\delta$ be the greatest power of 2 dividing any minor of order 2 of the submatrix $\left(\begin{smallmatrix}2 & \overset{g}{\cdots} & 2 \\\zeta_1 & \cdots & \zeta_g\end{smallmatrix}\right)$ of (4.1). These minors have the form $2(\zeta_i-\zeta_j)$ . If g is even, some $\zeta_i$ is even and thus $2(\zeta_i-\zeta_j)$ is even but not multiple of 4 in some cases, since there is also some $\zeta_j$ that is odd; therefore, $\delta=1$ if g is even. If g is odd, every $\zeta_i$ may be odd and $\delta$ may be greater than 1. If $k=0$ and $r_2>0$ , we obtain from (4.1) that the free-rank of $({\Lambda}/N)_{ab}$ is $g-2$ and

   \begin{equation*}Syl_2(\mathcal{T}(({\Lambda}/N)_{ab})) \approx {\mathbb{Z}}_{2^{{\widehat\mu}_1(2)}} \oplus \cdots \oplus {\mathbb{Z}}_{2^{{\widehat\mu}_{r-1}(2)}} \oplus {\mathbb{Z}}_{2^{{\widehat\mu}_r(2)+\delta}}\end{equation*}
   The claim for $2^{\alpha_i}$ and $2^{\alpha_{m+n-g+2}-1}$ follows from Lemma 3.2 when applied to the epimorphism $Syl_2(({\Lambda}/N)_{ab})\to A_2$ if $g<3$ and ${\mathbb{Z}}^{g-2}\oplus Syl_2(\mathcal{T}(({\Lambda}/N)_{ab}))$ $\to A_2$ otherwise.

   Now, suppose also that $g=2$ , $2^{\alpha_{m+n}}$ divides no proper period and $\alpha_{m+n-1}<\alpha_{m+n}-1$ . If $2^{\alpha_{m+n}-1}$ divides an even number of proper periods, then the last component of ${\theta}_2(x_1\cdots x_r)$ is doubly even. Also, the last component of either ${\theta}_2(d_1)$ or ${\theta}_2(d_2)$ is odd in order to generate ${\mathbb{Z}}_{\alpha_{m+n}}$ , but both last components cannot be odd (otherwise, every element in $\ker{\theta}_2$ would contain an even number of glide reflections and would be orientation-preserving) and thus the last component of ${\theta}_2(d_1^2d_2^2)$ is singly even. Therefore, the long relation would not be preserved, since the last component of ${\theta}_2(x_1\cdots x_rd_1^2d_2^2)$ would be singly even.

4. (iv) Otherwise, the component of ${\theta}_2(x_1\cdots x_r\cdot d_1^2\cdots d_g^2)$ corresponding to some factor of order $2^{\alpha_{m+n}}$ would be odd and the long relation would not be preserved.

5. (v) If $k=0$ and $r_2=0$ , the dimensions of the relation matrix (4.1) are $(r+2)\times(r+g)$ . Since $\zeta_i$ is odd for some $i\in\{1,\ldots,g\}$ and $m_i$ is odd, it follows that $\rho_i$ is odd for $i<r+2$ and $\rho_{r+2}$ is even, and thus the free-rank of $({\Lambda}/N)_{ab}$ is $g-2$ and its Sylow 2-subgroup is nontrivial cyclic. As an epimorphism $({\Lambda}/N)_{ab}\to A$ exists, we have $g-2+1\geqslant m+n$ by choosing $q=2$ and $j=1$ in (3.1), hence $g>m+n$ . So if, in addition, $n=0$ then $g\geqslant m+1$ ; but m odd and $g=m+1$ (even) would be inconsistent with (3.1): g even (and $k=0$ ) implies that some $\zeta_i$ is even ( $\sum \zeta_i$ is odd) and thus $\rho_{r+2}$ is even but not multiple of 4 (since $r_2=0$ ), so the Sylow 2-subgroup of $({\Lambda}/N)_{ab}$ is ${\mathbb{Z}}_2$ ( $\rho_i$ is odd for $i<r+2$ since $r_2=0$ and some $\zeta_i$ is odd) and we obtain $g-2\geqslant m$ for $q=2$ and $j=2$ in (3.1) and thus $g>m+1$ .

6. (vi) Let $\textbf{R}$ be the matrix (4.1), depending on the signature sign of ${\Lambda}$ . If $k>0$ , then either some $\kappa_i$ equals 1 or some $\zeta_i$ is odd, say, $\zeta_1$ (recall that $\sum\zeta_i+\sum\kappa_i$ is odd). Then, $\rho_2$ is odd since, in the $r+1$ -th and $r+2$ -th rows of $\textbf{R}$ , there is a $2\times 2$ -submatrix like $\left(\begin{smallmatrix}1 & 0 \\ \cdots & 1\end{smallmatrix}\right)$ in the first case, and like $\left(\begin{smallmatrix}2 & 1 \\ \zeta_1 & \cdots\end{smallmatrix}\right)$ in the second case. Also, $\rho_1=1$ since some entry equals 1. Therefore, $\epsilon_1=\rho_1=1$ and $\epsilon_2=\rho_2/\rho_1$ is odd in the Smith normal form of $\textbf{R}$ , so the free-rank of $({\Lambda}/N)_{ab}$ is $\eta g+k-2=w-1$ and its Sylow 2-subgroup has, at most, $r_2+S$ factors. By (3.1) applied for $q=2$ and $j=1$ to the epimorphism ${\mathbb{Z}}^{w-1}\oplus Syl_2(({\Lambda}/N)_{ab})\to A_2$ , the free-rank of $({\Lambda}/N)_{ab}$ and the number of factors of its Sylow 2-subgroup add up to $m+n$ (the number of factors of $A_2$ ) or more. Hence, $w-1+r_2+S\geqslant m+n$ .

7. (vii) This condition follows from Lemma 4.1.

8. (viii) If $Syl_2(A)\approx {\mathbb{Z}}_{2^{\alpha}}$ and $(n_{i1},\ldots,n_{is_i})=(2, 2, \ldots, 2)$ is a period cycle of ${\Lambda}$ , then ${\theta}(c_{ij-1})={\theta}(c_{ij})=2^{\alpha-1}$ , since both ${\theta}(c_{ij-1})$ and ${\theta}(c_{ij})$ are elements of order 2 in $Syl_2(A)$ . But then ${\theta}(c_{ij-1}c_{ij})={0}$ and ${\theta}(c_{ij-1}c_{ij})$ cannot have order $n_{ij}=2$ , in contradiction with Theorem [Reference Bujalance3, Proposition 3.2].

9. (ix) Assume that $Syl_2(A)\approx {\mathbb{Z}}_{2^{\alpha_1}}\oplus{\mathbb{Z}}_{2^{\alpha_2}}$ . By the relation $e_i^{-1}c_{i0}e_ic_{is_i}=1$ , ${\theta}(c_{i0})={\theta}(c_{is_i})$ , and thus $s_i$ is even, since ${\theta}(c_{ij-1})\neq{\theta}( c_{ij})$ (otherwise ${\theta}(c_{ij-1}c_{ij})=({0},{0})$ and ${\theta}(c_{ij-1}c_{ij})$ would not have order $n_{ij}=2$ ).

We prove the sufficiency of the conditions by defining epimorphisms ${\theta}_{q}\,:\,{\Lambda}\to A_{q}$ for each prime q in the set $\{q_1,\ldots,q_\lambda\}$ of prime numbers dividing the order of A, and a surface-kernel epimorphism ${\theta}\,:\,{\Lambda}\to A$ as the direct product epimorphism

\begin{equation*}{\theta} \,:\, {\Lambda} \to A \,:\, g \,\mapsto {\theta}(g) = ({\theta}_{q_1}(g), \ldots, {\theta}_{q_\lambda}(g)).\end{equation*}

For readability, we let $\mu_i=\mu_i(q)$ (see Section 3) in the definition of each homomorphism ${\theta}_q$ . Also, we assume that $\mu_i\leqslant\mu_{i+1}$ ; otherwise, there is a permutation (in general, different for each value of q) such that ${\widehat\mu}_i=\mu_{\tau(i)}$ and we replace $x_i$ by $x_{\tau(i)}$ and $\mu_i$ by ${\widehat\mu}_i$ in the definition of ${\theta}_q(x_i)$ below, so that the order of ${\theta}(x_i)$ is $m_i$ .

First, we define ${\theta}_2$ (whenever $m+n>0$ ) for $k=0$ (the cases not showed here are defined alike) and $k>0$ , and then we define ${\theta}_q$ for $q\neq 2$ . Let $\Sigma=\sum_i 2^{\alpha_{m+n}-\mu_i}$ (note that, if $\Sigma$ is odd, then $2^{\alpha_{m+n}}$ divides an odd number of proper periods; recall condition (iv) in that case).

1. a) $k=r_2=0$ and $n>0$ (so $g>m+n$ ).

   \begin{align*}&{\theta}_2(d_1) = ({1}, -1, \ldots, -1), {\theta}_2(d_i) = ({0}, \overset{i-1}{\ldots}, {0}, {1}, {0}, \ldots, {0}), \quad i=2,\ldots, m+n, \\&{\theta}_2(d_i) = ({0}, \ldots, {0}), \quad i>m+n, \quad d_g\in\ker{\theta}_2\cap({\Lambda}-{\Lambda}^+).\end{align*}

2. b) $k=r_2=n=0$ (so $g>m+1$ or $g=m+1$ is odd).

   \begin{align*}&{\theta}_2(d_1) = ({-}1, \ldots, -1), {\theta}_2(d_i) = ({0}, \overset{i-2}{\ldots}, {0}, {1}, {0}, \ldots, {0}), \quad i=2,\ldots, m+1, \\&{\theta}_2(d_i) = ({0}, \ldots, {0}), \quad i>m+1, \text{either $d_g$ or $d_1\cdots d_g\in\ker{\theta}_2\cap({\Lambda}-{\Lambda}^+)$.}\end{align*}

3. c) $k=0$ , $r_2>0$ and $m+n\leqslant g-2$ .

   \begin{align*}&{\theta}_2(x_i) = ({0}, \ldots, {0}, 2^{\alpha_{m+n}-\mu_i}), \quad i=1,\ldots, r, \\&{\theta}_2(d_i) = ({0}, \overset{i-1}{\ldots}, {0}, {1}, {0}, \ldots, {0}), \quad i=1,\ldots, m+n, \\&{\theta}_2(d_{m+n+1}) = ({-}1, \ldots, -1, -1-\Sigma/2), \\&{\theta}_2(d_i) = ({0}, \ldots, {0}), \quad i\geqslant m+n+2,\\&d_g\in\ker{\theta}_2\cap({\Lambda}-{\Lambda}^+).\end{align*}

4. d) $k=0$ , $r_2>0$ , $m+n>g-2$ and $g>1$ is odd.

   \begin{align*}&{\theta}_2(x_i) = ({0}, \ldots, {0}, 2^{\alpha_{m+n}-\mu_i}), \quad i=1,\ldots, r-m-n+g-2, \\&{\theta}_2(x_i) = ({0}, \ldots, {0}, {1}, {0}, \overset{g+r-i-3}{\ldots}, {0}, 2^{\alpha_{m+n}-\mu_i}), \\ &\qquad\qquad\qquad\qquad\qquad\qquad i=r-m-n+g-1,\ldots, r-2, \\&{\theta}_2(d_i) = ({0}, \ldots, {0}, {1}, {0}, \overset{g-i-1}{\ldots}, {0}), \quad i=1,\ldots, g-2.\end{align*}
   We consider the following cases:

   1. d.1) $\Sigma$ is singly even.

      \begin{align*}&{\theta}_2(x_{r-1}) = ({0}, \ldots, {0}, 1, {0}, \overset{g-2}{\ldots}, {0}, 2^{\alpha_{m+n}-\mu_{r-1}}), \\&{\theta}_2(x_r) = ({-}1, \ldots, -1, {0}, \overset{g-2}{\ldots}, {0}, 2^{\alpha_{m+n}-\mu_r}), \\&{\theta}_2(d_{g-1}) = ({0}, \ldots, {0}, -1, \overset{g-2}{\ldots}, -1, -\Sigma/2), \\&{\theta}_2(d_g) = ({0}, \ldots, {0}), \\&d_g\in\ker{\theta}_2\cap({\Lambda}-{\Lambda}^+).\end{align*}

   2. d.2) $\Sigma$ is doubly even.

      \begin{align*}&{\theta}_2(x_{r-1}) = ({0}, \ldots, {0}, 1, {0}, \overset{g-2}{\ldots}, {0}, 2^{\alpha_{m+n}-\mu_{r-1}}), \\&{\theta}_2(x_r) = ({-}1, \ldots, -1, {0}, \overset{g-2}{\ldots}, {0}, 2^{\alpha_{m+n}-\mu_r}), \\&{\theta}_2(d_{g-1}) = ({0}, \ldots, {0}, 1), \\&{\theta}_2(d_g) = ({0}, \ldots, {0}, -1, \overset{g-2}{\ldots}, -1, -1-\Sigma/2), \\&d_1\cdots d_{g-2}\cdot d_{g-1}^{1+\Sigma/2}\cdot d_g\in\ker{\theta}_2\cap({\Lambda}-{\Lambda}^+).\end{align*}

   3. d.3) $\Sigma$ is odd and $\Sigma-1$ is singly even.

      \begin{align*}&{\theta}_2(x_{r-1}) = ({0}, \ldots, {0}, 1, {0}, \overset{g-3}{\ldots}, {0}, {1}, {1}), \\&{\theta}_2(x_r) = ({-}1, \ldots, -1, {0}, \overset{g-3}{\ldots}, {0}, -{1}, {0}), \\&{\theta}_2(d_{g-1}) = ({0}, \ldots, {0}, -1, \overset{g-2}{\ldots}, -1, -(\Sigma-1)/2), \\&{\theta}_2(d_g) = ({0}, \ldots, {0}), \\&d_g\in\ker{\theta}_2\cap({\Lambda}-{\Lambda}^+) \\\end{align*}

   4. d.4) $\Sigma$ is odd and $\Sigma-1$ is doubly even.

      \begin{align*}&{\theta}_2(x_{r-1}) = ({0}, \ldots, {0}, 1, {0}, \overset{g-3}{\ldots}, {0}, {1}, {1}), \\&{\theta}_2(x_r) = ({-}1, \ldots, -1, {0}, \overset{g-3}{\ldots}, {0}, -{1}, {0}), \\&{\theta}_2(d_{g-1}) = ({0}, \ldots, {0}, 1), \\&{\theta}_2(d_g) = ({0}, \ldots, {0}, -1, \overset{g-2}{\ldots}, -1, -1-(\Sigma-1)/2), \\&d_1\cdots d_{g-2}\cdot d_{g-1}^{1+(\Sigma-1)/2}\cdot d_g\in\ker{\theta}_2\cap({\Lambda}-{\Lambda}^+).\end{align*}

When $k>0$ , we define ${\theta}_2(c_{10})=({0},\ldots,{0},2^{\alpha_{m+n}-1})$ and consider the sequence

\begin{equation*}(c_{20}, \ldots, c_{{\varepsilon}0}, c_{{\varepsilon}+1, 0}, \ldots, c_{{\varepsilon}+1, s_{{\varepsilon}+1}-1}, \ldots, c_{k0}, \ldots, c_{k, s_k-1})\end{equation*}

containing $S-1$ elements—we rule out the elements $c_{10}$ and $c_{is_i}$ for $i>{\varepsilon}$ . We subsequently assign $(2^{\alpha_{1}-1}, {0},\ldots, {0})$ , $({0}, 2^{\alpha_{2}-1}, {0},\ldots, {0})$ , etc—beginning again with $(2^{\alpha_{1}-1}, {0},\ldots, {0})$ if there are more than $m+n$ elements in the sequence. Also, we define ${\theta}_2(c_{is_i})={\theta}_2(c_{i0})$ for $i>{\varepsilon}$ .

We consider the sequence $a_g,b_g\ldots,a_1,b_1,e_{k-1},\ldots,e_1,x_r,\ldots,x_1$ or $d_g,\ldots, d_1,e_{k-1},\ldots,e_1,$ $x_r,\ldots,x_1$ according to the signature of ${\Lambda}$ and subsequently assign $({0},\ldots,{0},{1}),({0},\ldots,{0},{1},{0}), \ldots,$ $ ({0},\overset{\min(n,S-1)}{\ldots\ldots},{0},{1},{0},\ldots,{0})$ to its first $m+n-\min(n,$ $S-1)$ elements and $({0},\ldots,{0})$ to the rest, but we let the last component be $(\ldots,2^{\alpha_{m+n}-\mu_i})$ in case of an elliptic generator $x_i$ ; also, let ${\theta}_2(d_g)=({0},\ldots,{0})$ if $m=0$ . Finally, let

\begin{equation*}{\theta}_2(e_k)=({0},\overset{\min(n,S-1)}{\ldots\ldots},{0},-1,\ldots,-1,-2,\overset{\eta g-1}{\ldots\ldots},-2,\delta-\Sigma),\end{equation*}

where $\delta =-1$ if $g=0$ , $\delta =0$ if $g>0$ and sign $({\Lambda})$ is ‘ $+$ ’, and $\delta =-2$ otherwise. We observe that $d_g$ (if $m=0$ ), $c_{10}d_g^p$ (for some even integer p) or $c_{10}a_g^p$ or $c_{10}e_{k-1}^p$ (for some integer p) belong to $\ker{{\theta}_2}\cap ({\Lambda}-{\Lambda}^+)$ .

With the above definition for ${\theta}_2$ , we can easily find, for each component of $A_2$ , an element in ${\Lambda}$ whose image is a generator (an odd number) of that component and null for the other components, so that ${\theta}_2$ is onto.

Now, let $q\neq 2$ be a prime number dividing $|A|$ and $A_q \approx {\mathbb{Z}}_{q^{\alpha_1}} \oplus \cdots \oplus {\mathbb{Z}}_{q^{\alpha_{t}}}$ be the q-Sylow subgroup of A (some factors of $A_q$ may be trivial, i.e., $\alpha_1=\cdots=\alpha_{t'}=0$ for some $t'<t$ ). Let

\begin{equation*}\gamma_1=e_1, \;\ldots, \;\gamma_{k-1}=e_{k-1}, \;\gamma_{k}=a_{1}, \;\gamma_{k+1}=b_{1}, \;\ldots, \;\gamma_{w-1}=a_{g}, \;\gamma_{w}=b_{g},\end{equation*}

or

\begin{equation*}\gamma_1=e_1, \ldots, \gamma_{k-1}=e_{k-1}, \gamma_{k}=d_{1}, \ldots , \gamma_{w}=d_{g},\end{equation*}

according to the sign of the signature of ${\Lambda}$ . We define ${\theta}_q$ as follows (note that $r+{w}\geqslant t$ by condition (ii)):

\begin{align*}{\theta}_q(c_{i0}) &= ({0}, \ldots, {0}), \qquad\qquad \qquad \qquad \quad i=1,\ldots, k, \\{\theta}_q(x_i) &= ({0}, \ldots, {0}, q^{\alpha_t-\mu_i}), \qquad\qquad\qquad\, i=\begin{cases} 1,\ldots, r-t+{w} & \text{if $t>{w}$}, \\ 1,\ldots, r & \text{if $t\leqslant {w}$},\end{cases} \\{\theta}_q(x_i) &= ({0}, \overset{t-r-{w}+i-1}{\ldots\ldots\ldots\ldots}, {0}, {1}, {0}, \overset{r+{w}-i-1}{\ldots\ldots\ldots}, {0}, q^{\alpha_{t}-\mu_i}), \\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\, i=r-t+{w}+1,\ldots, r \quad\text{if $t>{w}$}, \\{\theta}_q(\gamma_i) &= ({0}, \ldots, {0}), \qquad\qquad\qquad\qquad\qquad\quad i=1,\ldots, {w}-t \quad\text{if $t<{w}$}, \\{\theta}_q(\gamma_i) &= ({0}, \overset{t-{w}+i-1}{\ldots\ldots\ldots}, {0}, {1}, {0}, \overset{{w}-i}{\ldots\ldots}, {0}), \quad i=\begin{cases} 1,\ldots, {w} & \text{if $t\geqslant {w}$}, \\ {w}-t+1,\ldots, {w} & \text{if $t< {w}$},\end{cases}\end{align*}

\begin{equation*}{\theta}_q(e_{k}) = \begin{cases}({-}1, \ldots, -1, \delta, \overset{\eta g-1}{\ldots\ldots}, \delta, -u+\delta ) & \text{if $t>\eta g>0$}, \\(\delta, \ldots, \delta, -u+\delta ) & \text{if $t\leqslant\eta g$ or $g=0$},\end{cases}\end{equation*}

where $u=\sum_{i=1}^{r} q^{\alpha_t-\mu_i}$ . The homomorphism ${\theta}_q$ is onto by condition (ii).

The homomorphism ${\theta}$ is also onto. For, consider an elementary divisor $q^{\alpha_i(q)}$ of A and the generator $h=({0}, \ldots, {0}, {1}, {0}, \ldots, {0})$ of some cyclic factor

\begin{equation*}H=\{{0}\} \oplus \cdots \oplus \{{0}\} \oplus{\mathbb{Z}}_{q^{\alpha_i(q)}} \oplus \{{0}\} \oplus \cdots \oplus \{{0}\}\end{equation*}

of $A_q$ . Then, $h={\theta}_q(g)$ for some $g\in{\Lambda}$ . Obviously, ${\theta}(g)$ may have nontrivial components in some other primary component $A_{q'}$ for a prime $q'\neq q$ , but not the element $\frac{\exp A}{q^{\alpha_t(q)}} \, {\theta}(g)$ since $\frac{\exp A}{q^{\alpha_t(q)}} \, {\theta}_{q'}(g)$ is trivial whenever $q'\neq q$ . Moreover, $\frac{\exp A}{q^{\alpha_t(q)}} \, {\theta}(g)$ has order $q^{\alpha_i(q)}$ since $\gcd(q, \exp A/q^{\alpha_t(q)})=1$ . Hence, $\langle{\theta}(g^{\exp A/q^{\alpha_t(q)}})\rangle = H$ .

It also preserves the long relation and the order of $x_i$ , $c_{ij}$ and $c_{ij-1} c_{ij}$ . We have emphasized an element in $\ker{{\theta}_2}\cap ({\Lambda}-{\Lambda}^+)$ , say h, for each case in the definition of ${\theta}_2$ . Since $v_t$ is odd (let $v_t=1$ if $t=0$ ), the element $h^{v_t}$ belongs to $\ker{{\theta}}\cap ({\Lambda}-{\Lambda}^+)$ and thus ${\theta}({\Lambda}^+)=A$ .

When A is cyclic, Theorem 4.3 reads as follows.

This result gathers theorems 3.5, 3.6 and 3.7 in [Reference Bujalance3], yet it rounds out Theorem 3.6 with a complete set of necessary and sufficient conditions.

5. Symmetric cross-cap number of an Abelian group

In this section, we develop a new way of achieving the expression for the symmetric cross-cap number of an Abelian group by means of Theorem 4.3.

The symmetric cross-cap number of a cyclic group was obtained by Bujalance in [Reference Bujalance3] from the above mentioned theorems 3.5, 3.6 and 3.7, so that, by the same token, it can also be obtained from Theorem 4.4 herein:

Theorem 5.1.

(5.1) \begin{equation}\widetilde{\sigma}({\mathbb{Z}}_N) = \begin{cases}3 & \text{if $N=2$,}\\[4pt](q-1)\left(N/q-1\right)+1 & \text{if} {N} \text{is not prime and $q^2\nmid N$,} \\[4pt](q-1)\left(N/q-1\right)+q & \text{otherwise,}\end{cases}\end{equation}

where q is the smallest prime divisor of N.

Remark 5.2. The symmetric cross-cap number of the groups ${\mathbb{Z}}_2^{\,2}$ , ${\mathbb{Z}}_2^{\,3}$ and ${\mathbb{Z}}_2\oplus{\mathbb{Z}}_{2u}$ ( $u>1$ ) was obtained in [Reference Gromadzki9, Proposition 6.4]:

(5.2) \begin{equation}\begin{gathered}\widetilde{\sigma}({\mathbb{Z}}_2^{\,2}) = 3, \quad\widetilde{\sigma}({\mathbb{Z}}_2^{\,3}) = 4, \quad\widetilde{\sigma}({\mathbb{Z}}_2\oplus{\mathbb{Z}}_{2u}) = 2u,\end{gathered}\end{equation}

By Theorem 4.3, we can obtain signatures of NEC groups attaining such topological genera: $(0;\,+;\,[{-}];\,\{(2,2,2)\})$ , $(0;\,+;\,[{-}];$ $\{(2,2,2,2)\})$ and $(0;\,$ $+;$ $[2,2u];$ $\{({-})\})$ , respectively, fulfill conditions of Theorem 4.3 and it can be proved that any other signature fulfilling such conditions yields a greater or equal topological genus.

Theorem 5.3. [Reference Gromadzki9, propositions 6.1, 6.2 and 6.3] The symmetric cross-cap number of a noncyclic Abelian group A different to ${\mathbb{Z}}_2^{\,2}$ , ${\mathbb{Z}}_2^{\,3}$ and ${\mathbb{Z}}_2\oplus{\mathbb{Z}}_{2u}$ ( $u\geqslant 2$ ) is $\widetilde{\sigma}(A)=2 + |A|\,\mu^*$ , where $\mu^*$ is, with the notation of Theorem 4.3,

\begin{align*}& \text{{a)} } t-1-\frac{1}{v_1}-\cdots-\frac{1}{v_{t-n}} && \text{if $n\leqslant m$,}\\&\text{{b)} } t-1-\frac{1}{v_1}-\cdots-\frac{1}{v_{t-\epsilon}}+\frac{\delta}{2v_{t-\epsilon}} && \text{if $2m< m+n< 2t$,} \\[3mm]& \text{{c)} } t-1 && \text{if $2m<m+n=2t$,} \notag\\[1mm]& \text{{d)} } t-1+\frac{m+n-2t+1}{4} && \text{if $2t<m+n$,} \notag\end{align*}

$\epsilon=(m+n-\delta)/2$ , $\delta=1$ if $m+n$ is odd and $\delta=0$ otherwise.

Proof. We first tackle odd order Abelian groups, so that $n=m=0$ and we only consider case a), for which $\mu^*=-1 + \sum_{i=1}^{t} \left( 1-\frac{1}{{v}_i} \right)$ . For, note that the signature $(1;\,-;\,[{v}_1,\ldots, {v}_t];\,\{-\})$ defines an NEC group ${\Lambda}^*$ and fulfills conditions of Theorem 4.3. Now we prove that, if ${\Lambda}$ is another NEC group with signature $(g;\,-;\,[m_1,$ $\ldots, m_r];$ $\{-\})$ fulfilling conditions of Theorem 4.3, then $\mu({\Lambda}^*)\leqslant \mu({\Lambda})=g-2 + \sum_{i=1}^{r} \left( 1-\frac{1}{m_i} \right)$ .

If $t\leqslant g-1$ , then $\mu({\Lambda}^*)<-1+t < -1+t + \sum_{i=1}^{r} \left( 1-\frac{1}{m_i} \right) \leqslant g-2 + \sum_{i=1}^{r} \left( 1-\frac{1}{m_i} \right) = \mu({\Lambda})$ . Suppose now that $t>g-1$ . By (2.4), we may assume that $m_1|\cdots|m_r$ , since $\mu({\widehat\Lambda})\leqslant \mu({\Lambda})$ if ${\widehat\Lambda}$ is an NEC group with signature $(g;\,-; [{\widehat m}_{r-{\widehat r}+1}, \ldots, {\widehat m}_{r}];\,\{-\})$ . By condition (ii) of Theorem 4.3, ${v}_1\,|\,m_{r+g-t}$ , $\ldots$ , ${v}_{t-g+1}\,|\,m_{r}$ , hence $\sum_{i=1}^{t-g+1} \left( 1-\frac{1}{{v}_i} \right) \leqslant \sum_{i=r-t+g}^{r} \left( 1-\frac{1}{m_i} \right)$ $\leqslant \sum_{i=1}^{r} \left( 1-\frac{1}{m_i} \right)$ (recall that $r-t+g\geqslant 1$ by condition (ii) of Theorem 4.3). It follows that $\mu({\Lambda}^*)\leqslant \mu({\Lambda})$ if $g=1$ , and also if $g>1$ since $\sum_{i=t-g+2}^{t} \left( 1-\frac{1}{{v}_i} \right) < \sum_{i=t-g+2}^{t} 1 =g-1$ .

Now, we consider Abelian groups of even order. Let ${\Lambda}^*$ be an NEC group with signature

\begin{align*}a) &\quad (0;\,+;\,[v_1,\ldots,v_{t-n}];\,\{({-})^{n+1}\}) \\b) &\quad (0;\,+;\,[v_1,\ldots, v_{t-\frac{m+n-\delta}{2}-1}, (\delta+1)v_{t-\frac{m+n-\delta}{2}}];\,\{({-})^{\frac{m+n-\delta}{2}+1}\}) \\c) &\quad (0;\,+;\,[{-}];\,\{({-})^{t+1}\}) \\d) &\quad (0;\,+;\,[{-}];\,\{({-})^{t}, (2,\overset{m+n-2t+1}{\ldots},2)\})\end{align*}

respectively. This NEC group fulfills conditions of Theorem 4.3 and $\mu({\Lambda}^*)=\mu^*$ . Now, we prove that it follows from Theorem 4.3 that $\mu^*\leqslant\mu({\Lambda})$ for any other NEC group ${\Lambda}$ fulfilling such conditions.

We can assume that ${\Lambda}$ has signature

(5.3) \begin{align}(0;\,+;\,[m_1,\ldots, m_r];\,\{({-})^{k-1}, (2,\overset{s}{\ldots},2)\}),\end{align}

where $m_i|m_{i+1}$ , $k>0$ , $s\neq 1$ , s is even if $m+n=1$ and $s=0$ if $m+n=0$ . For, the signature $(g;\,\pm; [{\widehat m}_{r-{\widehat r}+1}, \ldots, {\widehat m}_{r}];\,\{({-})^{\varepsilon}, (2,\overset{s_{\varepsilon+1}}{\ldots},2), \ldots, (2,\overset{s_k}{\ldots},2)\})$ defines an NEC group ${\widehat\Lambda}$ that fulfills conditions of Theorem 4.3 and, by (2.4), $\mu({\widehat\Lambda})\leqslant\mu({\Lambda})$ if the signature of ${\Lambda}$ is $(g;\,\pm;$ $[m_1,\ldots, m_r];\,\{({-})^{\varepsilon}, (2,\overset{s_{\varepsilon+1}}{\ldots},2), \ldots, (2,\overset{s_k}{\ldots},2)\})$ also fulfilling such conditions (we can easily see that also $\mu({\widehat\Lambda})>0$ in that case). Therefore, we can assume that $m_1|\cdots|m_r$ . Moreover, consider an NEC group ${\Lambda}_o$ with signature $(0;\,+;\,[m_1,\ldots, m_r];$ $\{({-})^{\eta g+k-1},$ $(2,\overset{s}{\ldots},2)\}),$ where $s=\sum_{i=1}^k s_i$ and $\eta, g$ and k are the parameters of ${\Lambda}$ (note that ${\Lambda}_o$ has $\eta g+k$ period-cycles, and thus it is a proper NEC group: $\eta g+k-1\geqslant 0$ since ${\Lambda}$ is proper). It is straightforward to check that $\mu({\Lambda}_o) = \mu({\Lambda})$ and ${\Lambda}_o$ fulfills the conditions of Theorem 4.3 if ${\Lambda}$ does.

So assume that ${\Lambda}$ has signature (5.3) and let $w=k-1$ , $\mu = \mu({\Lambda}) = w-1 + \sum_{i=1}^r$ $\left(1-\frac{1}{m_i}\right) + \frac{s}{4}$ and $w^*=n$ , $(m+n-\delta)/2$ , t and t for cases a), b), c) and d) in the definition of $\mu^*$ in Theorem 5.3, respectively.

If $t\leqslant w$ , then $\mu^* \leqslant t-1 \leqslant w-1 \leqslant \mu$ for cases a), b) and c). For case d), if $m+n-2w-s+1\leqslant 0$ , then $\mu \geqslant w-1+\frac{s}{4} \geqslant w-1+ \frac{m+n-2w+1}{4} = \mu^* + \frac{w-t}{2} \geqslant \mu^*,$ and, if $m+n-2w-s+1 > 0$ , then $\sum (1-1/m_i) \geqslant (m+n-2w-s+1)/2$ by condition (vi) and $\mu \geqslant w-1+ \frac{m+n-2w-s+1}{2}+ \frac{s}{4} = \mu^* + \frac{m+n-2t-s+1}{4} \geqslant \mu^*$ .

Otherwise, $t>w$ . If $t>w>w^*$ (that discards cases c) and d)), then $\mu^* \;<\; w^*-1 + \sum_{i=1}^{t-w} \left(1-\frac{1}{v_i}\right) + \sum_{i=t-w+1}^{t-w^*} 1 \;=\; w-1 + \sum_{i=1}^{t-w} \left(1-\frac{1}{v_i}\right) \;\leqslant\; \mu$ since $v_1|m_{r-t+w+1}, \ldots, v_{t-w}|m_r$ by condition (ii).

If $t>w=w^*$ (we discard cases c) and d) as well), then, in cases a) and b) with $m+n$ even ( $\delta=0$ ), $\mu^* \;=\; w-1 + \sum_{i=1}^{t-w} \left(1-\frac{1}{v_i}\right) \;\leqslant\; \mu$ by condition (ii) as above. In case b) with $m+n$ odd ( $\delta=1$ ), $1-\frac{1}{2v_{t-w}} = 1-\frac{1}{v_{t-w}} +\frac{1}{2v_{t-w}} \leqslant 1-\frac{1}{m_r}+\frac{s}{4} $ provided that $s\geqslant 2$ , and, if $s=0$ , then there is, at least, $m+n-2w=m+n-(m+n-1)=1$ even proper period by condition (vi) (note that $S-1=w$ if $s=0$ ) and thus $2v_{t-w}|m_r$ since $v_{t-w}$ is odd (note that $t-w\leqslant t-m$ since, in case b), $w^*\geqslant m$ ). Therefore, $\mu^*\leqslant\mu$ either if $s>0$ or $s=0$ .

Finally, suppose that $t>w$ and $w^*>w$ . We proof that $\mu^*<\mu$ for case a) (cases b), c) and d) are established similarly). To lessen clutter, we rename the first $t-w$ integers $v_i$ by defining $v^{\prime}_1=\cdots=v^{\prime}_{r-t+w}=1$ , $v^{\prime}_{r-t+w+1}=v_1$ , $\ldots$ , $v^{\prime}_{r}=v_{t-w}$ . Hence, $v^{\prime}_i|m_i$ for all i by condition (ii), and thus $1-\frac{1}{v^{\prime}_i}\leqslant 1-\frac{1}{m_i}$ .

For case a) we have $t\geqslant m\geqslant n=w^*>w$ . We consider the following partition of $\{1,\ldots,r\}$ : ( $\#A<t-m$ in the figure, but $\#A$ may be greater than $t-m$ ). Let $A=\varnothing$ if $n-w-s+1\leqslant 0$ . In case that $s=0$ , let $\#A=n-w$ .

Note that $4|v^{\prime}_i$ if $i\in B\cup C$ and $v^{\prime}_i$ is odd otherwise, and $\mu^*=n-1+\sum_{i\not\in C} (1-\frac{1}{v^{\prime}_i})$ .

If $w+s>n$ (hence $s\geqslant 2$ since $w<n$ ), then $\sum_{C} \left(1-\frac{1}{m_i}\right) + \frac{s}{4} > \frac{3(n-w)}{4} + \frac{n-w}{4}= n-w$ since $4|m_i$ for $i\in C$ , and thus $\mu>w-1+\sum_{i\not\in C} (1-\frac{1}{m_i})+n-w\geqslant \mu$ .

If $w+s-1 < n$ and $s\geqslant 2$ , then $m+n-2w-s+1>0$ since $m\geqslant n>w$ , and, by condition (vi), $m_i$ is even if $i\in A\cup B\cup C$ and thus $r\geqslant m+n-2w-s+1=\#\{A\cup B\cup C\}$ . Let $C=C_1\cup C_2$ , with $C_1=\{r-n+w+1,\ldots,r-s+1\}$ and $C_2=\{r-s+2,\ldots,r\}$ , $\#C_1=\#A$ , $\#C_2=s-1$ . For $i\in A$ , $v^{\prime}_i$ is odd and $m_i$ is even, hence $2v^{\prime}_i|m_i$ . Also, $4|m_j$ for $j\in C_1$ , hence $4v^{\prime}_i|m_j$ if $i\in A$ . Then $\sum_{A} \left(1-\frac{1}{m_i}\right) + \sum_{C_1} \left(1-\frac{1}{m_i}\right) \geqslant \sum_{A} \left(1-\frac{1}{2v^{\prime}_i}\right) + \sum_{A} \left(1-\frac{1}{4v^{\prime}_i}\right)$ $= n-w-s+1 + \sum_{A} \left(1-\frac{1}{2v^{\prime}_i}-\frac{1}{4v^{\prime}_i}\right) > n-w-s+1 + \sum_{A} \left(1-\frac{1}{v^{\prime}_i}\right)$ . As $4|m_i$ for $i\in C_2$ and $\#C_2=s-1$ , $\sum_{C_2} \left(1-\frac{1}{m_i}\right) + \frac{s}{4} \geqslant (s-1) \left(1-\frac{1}{4}\right) + \frac{s}{4} = s-1 + \frac{1}{4} \geqslant s-1$ .

If $w+s-1 < n$ and $s=0$ , then $C_2=\varnothing$ , $\#A=\#C=n-w$ and $\sum_{A\cup C} \left(1-\frac{1}{m_i}\right)$ $> n-w + \sum_{A} \left(1-\frac{1}{v^{\prime}_i}\right)$ . It follows that $\mu^* \leqslant \mu$ if either $s\geqslant 2$ or $s=0$ .

6. Least symmetric cross-cap number of Abelian groups of a given order

An easy consequence of the results of the previous section is the following: for a given integer $N>1$ , we find the least topological genus of any nonorientable Riemann surface of topological genus $g>2$ on which some Abelian group of order N acts. For ease and by abuse of notation, we denote it by $\widetilde{\sigma}(N)$ (it is not the symmetric cross-cap number of a group but the least symmetric cross-cap number attained in a family of groups).

Theorem 6.1. The least symmetric cross-cap number of Abelian groups of order $N>1$ is

\begin{equation*}\widetilde{\sigma}(N)=\begin{cases}3 & \text{if $N\leqslant 4$,} \\[3pt]6 & \text{if $N=16$,} \\[3pt]N & \text{if $N>4$ is prime, and} \\[3pt](q-1)\left(N/q-1\right)+1 & \text{otherwise,}\end{cases}\end{equation*}

where q is the smallest prime divisor of N.

7. Maximum order problem

The maximum order problem for Abelian groups acting on Riemann surfaces of genus $g>1$ was solved in [Reference Breuer2, Corollary 9.6], and in [Reference Bujalance, Etayo, Gamboa and Gromadzki8, Section 4.5] for Abelian groups acting on compact bordered Klein surfaces of algebraic genus $p>1$ . We now obtain the corresponding result for compact nonorientable Riemann surfaces, which expands that of Bujalance for cyclic groups [Reference Bujalance3, Corollary 4.4]. It follows easily from theorems 4.3 and 6.1.

Proof. If $g=6$ , then the largest order is 16 since, by Theorem 6.1, $\widetilde{\sigma}(16)=6$ and $\widetilde{\sigma}(N)\geqslant N/2>6$ if $N>16$ (note that $(q-1)(N/q-1)+1=N/2+(q-2)(N-2q)/2q\geqslant N/2$ if $q\geqslant 2$ ).

If $g=8$ , then the largest order is 16 since $\widetilde{\sigma}({\mathbb{Z}}_2\oplus{\mathbb{Z}}_8)=8$ and $\widetilde{\sigma}(N)\geqslant N/2>8$ if $N>16$ .

Otherwise, $g\neq 6$ or 8. In that case, $\widetilde{\sigma}(2g)=g$ and thus the largest order is, at least, 2g. But no Abelian group of order greater than 2g acts on compact nonorientable Riemann surfaces of topological genus g. For, consider an Abelian group A of order N that acts on genus $g\neq 6$ or 8, so $g\geqslant\widetilde{\sigma}(N)$ . If $N\neq 16$ , then $\widetilde{\sigma}(N)\geqslant N/2$ by Theorem 6.1 and thus $N\leqslant 2g$ . Now, suppose that $N=16$ , hence $g\geqslant\widetilde{\sigma}(16)=6$ . If $g>8$ , then $16<2g$ . Finally, no Abelian group of order 16 acts on genus 7; indeed, $\widetilde{\sigma}(A)>7$ for any Abelian group A of order 16 other than ${\mathbb{Z}}_2^{\,4}$ and, if $A\approx{\mathbb{Z}}_2^{\,4}$ acts on genus 7, then $5/4=4w-4+2r+s$ for some nonnegative integers w, r, s by Theorem 4.3 and the Riemann-Hurwitz formula (2.3), which is not possible since $4w-4+2r+s\in{\mathbb{Z}}$ .