1 Introduction
There is a famous conjecture known as the noninner automorphism conjecture, listed in the book ‘Unsolved Problems in Group Theory: The Kourovka Notebook’, which states that every finite nonabelian p-group admits an automorphism of order p which is not an inner automorphism [Reference Khukhro and Mazurov11, Problem 4.13].
A sharpened version of the conjecture states that every finite nonabelian p-group G has a noninner automorphism of order p which fixes the Frattini subgroup
$\Phi (G)$
element-wise. This conjecture was first attacked by Liebeck [Reference Liebeck12]. He proved that for an odd prime p, every finite p-group G of nilpotency class
$2$
has a noninner automorphism of order p fixing
$\Phi (G)$
element-wise. For 2-groups of class 2, the conjecture was settled by Abdollahi [Reference Abdollahi1] in 2007. In 2013, Abdollahi et al. [Reference Abdollahi, Ghoraishi and Wilkens3, Theorem 4.4] proved that every finite p-group G of odd order and of nilpotency class 3 has a noninner automorphism of order p that fixes
$\Phi (G)$
element-wise. In 2013, Shabani-Attar [Reference Shabani-Attar15] proved that if G is a finite nonabelian p-group of order
$p^m$
and exponent
$p^{m-2}$
, then G has a noninner automorphism of order p. In 2014, Abdollahi et al. [Reference Abdollahi, Ghoraishi, Guerboussa, Reguiat and Wilkens4] showed that every finite p-group G of co-class 2 has a noninner automorphism of order p leaving
$Z(G)$
element-wise fixed. For more such results, see [Reference Deaconescu and Silberberg5–Reference Ghoraishi10].
If there is a maximal subgroup M of a finite p-group G with
$|G|> p$
and
${Z(M) \subseteq Z(G)}$
, then there exists a noninner automorphism of G of order p (see [Reference Rotman13, Lemma 9.108]). In 2002, Deaconescu and Silberberg [Reference Deaconescu and Silberberg5] proved that if G is a finite nonabelian p-group such that
$C_G(Z(\Phi (G))) \neq \Phi (G)$
, then G has a noninner automorphism of order p which fixes
$\Phi (G)$
element-wise. This reduces the verification of the conjecture to the special case in which
$C_G(Z(\Phi (G)))=\Phi (G)$
. If the conjecture is false for a finite p-group G, then it follows from [Reference Deaconescu and Silberberg5, Remark 2] that
$Z(G) < Z(M)$
for all maximal subgroups M of G. This suggests the following natural question.
Question 1.1. Given a finite p-group G with
$Z(G)<Z(M)$
for all maximal subgroups M of G, can the conjecture hold?
In Theorem 2.1, we prove that every finite p-group G (
$p>2$
) of nilpotency class n with
$\text {exp}(\gamma _{n-1}(G))=p$
,
$|\gamma _n(G)| =p$
and
$Z(C_G(x)) \le \gamma _{n-1}(G)$
for all
$x \in \gamma _{n-1}(G) \setminus Z(G)$
, has a noninner automorphism of order p which fixes
$\Phi (G)$
element-wise. As a consequence, in Corollaries 2.2 and 2.3, we give an affirmative answer to the above question under some conditions. In 2017, Ruscitti et al. [Reference Ruscitti, Legarreta and Yadav14] confirmed the conjecture for finite p-groups of co-class 3 with
$p \neq 3$
. We also validate the conjecture for some nonabelian finite 3-groups of co-class 3 in Corollary 2.4. In [Reference Ghoraishi8, Theorem 1.1], Ghoraishi improved the reduction given by Deaconescu and Silberberg [Reference Deaconescu and Silberberg5] by proving that if a finite nonabelian p-group G fails to fulfil the condition
$Z_2^{\star }(G) \le C_G(Z_2^{\star }(G)) = \Phi (G)$
, where
$Z_2^{\star }(G)$
is the pre-image of
$\Omega _1(Z_2(G)/Z(G))$
in G, then G has a noninner automorphism of order p which fixes
$\Phi (G)$
element-wise. At the end of Section 2, we provide an elementary and short proof of this result.
Throughout, p denotes a prime number. For a group G, by
$Z_m(G)$
,
$\gamma _{m}(G)$
,
$d(G)$
and
$\Phi (G)$
, we denote the mth term of the upper central series of G, the mth term of the lower central series of G, the minimum number of generators of G and the Frattini subgroup of G, respectively. The nilpotency class and the exponent of a finite group G are denoted by
$\text {cl}(G)$
and
$\text {exp}(G)$
, respectively. A finite p-group G of order
$p^n$
with
$\text {cl}(G)=n-c$
is said to be of co-class c. For a finite p-group G, we write
$\Omega _1(G)=\langle g \in G ~|~ g^p=1 \rangle $
. All other unexplained notation, if any, are standard.
2 Main results
Since the conjecture is true for all finite p-groups G having nilpotency class 2 and 3, we consider only finite p-groups G with
$\text {cl}(G) \ge 4$
.
Theorem 2.1. Let G be a finite p-group (
$p>2$
) of class n such that
$|\gamma _{n}(G)| =\text {exp}(\gamma _{n-1}(G))=p$
and
$Z(C_G(x)) \le \gamma _{n-1}(G)$
for all
$x \in \gamma _{n-1}(G) \setminus Z(G)$
. Then, G has a noninner automorphism of order p that fixes
$\Phi (G)$
element-wise.
Proof. Since
$n=\text {cl}(G) \ge 4$
and
$\text {exp}(\gamma _{n-1}(G))=p$
, there exists an element
${x \in \gamma _{n-1}(G)\setminus Z(G)}$
of order p. Thus,
$[x,G]\subseteq \gamma _n(G)$
and, therefore, the order of the conjugacy class of x in G is p. It follows that
$M=C_{G}(x)$
is a maximal subgroup of G. Let
$g\in G\setminus M$
. Then,
$$ \begin{align*} (gx)^{p}=g^{p}x^{p}[x,g]^{p(p-1)/2} =g^{p}. \end{align*} $$
Consider the map
$\beta $
of G defined by
$\beta (g)=gx$
and
$\beta (m)=m$
for all
$m\in M$
. The map
$\beta $
can be extended to an automorphism of G fixing
$\Phi (G)$
element-wise and of order p. We claim that
$\beta $
is a noninner automorphism of G. For a contradiction, assume that
$\beta =\theta _y$
, the inner automorphism of G induced by some
$y \in G$
, which implies that
$y\in C_G(M)$
. If
$y\notin M,$
then
$G=M\langle y\rangle $
. It follows that
$y \in Z(G)$
, which is a contradiction. Therefore,
$y\in Z(M)$
. Since
$\beta =\theta _y$
, we have
$g^{-1}\theta _y(g)=[g,y]=x$
. Now, by the hypothesis that
$Z(C_G(x)) \le \gamma _{n-1}(G)$
for all
$x \in \gamma _{n-1}(G) \setminus Z(G)$
, we have
$y \in \gamma _{n-1}(G)$
. Therefore,
$$ \begin{align*} x=[g,y] \in \gamma_n(G) \le Z(G), \end{align*} $$
which contradicts our choice of x in G. Hence, G has a noninner automorphism of order p that fixes
$\Phi (G)$
element-wise.
Let G be a finite p-group such that
$|Z(G)| =p$
. Let M be any maximal subgroup of G. Since
$Z(M)$
is a characteristic subgroup of M and M is a normal subgroup of G, it follows that
$Z(M)$
is a normal subgroup of G. Thus,
$Z(G)\le Z(M)$
for all maximal subgroups M of G. We obtain the following corollary from Theorem 2.1.
Corollary 2.2. Let G be a finite p-group (
$p>2$
) of class n such that
$|Z(G)| =\text {exp}(\gamma _{n-1}(G))=p$
and
$Z(C_G(x)) \le \gamma _{n-1}(G)$
for all
$x \in \gamma _{n-1}(G) \setminus Z(G)$
. Then, G has a noninner automorphism of order p that fixes
$\Phi (G)$
element-wise.
Corollary 2.3. Let G be a finite p-group (
$p>2$
) of class n such that
$|Z(G)|=p$
and
$Z(M)= \gamma _{n-1}(G)$
is of exponent p for all maximal subgroups M of G. Then, G has a noninner automorphism of order p that fixes
$\Phi (G)$
element-wise.
Proof. Given that
$\text {cl}(G)=n$
. It follows that
$\gamma _n(G) \le Z(G)$
. Consequently,
$|\gamma _n(G)| =p$
. Considering the hypothesis that
$Z(M)= \gamma _{n-1}(G)$
is of exponent p for all maximal subgroups M of G and the proof of Theorem 2.1, we deduce that
$Z(C_G(x)) \le \gamma _{n-1}(G)$
for all
$x \in \gamma _{n-1}(G) \setminus Z(G)$
. Hence, it follows from Theorem 2.1 that G possesses a noninner automorphism of order p that fixes
$\Phi (G)$
element-wise.
Corollary 2.4. Let G be a finite
$3$
-group of order
$3^n$
and of co-class
$3$
such that
$Z(M)=\gamma _{n-4}(G)$
is of exponent
$3$
for all maximal subgroups M of G. Then, G has a noninner automorphism of order
$3$
.
Proof. Assume that G does not possess any noninner automorphism of order 3. Then, it follows from [Reference Abdollahi2, Corollary 2.3] that
$$ \begin{align} d(Z_2(G)/Z(G))=d(Z(G))\,d(G). \end{align} $$
Since G is of co-class 3, we have
$p^i \le |Z_i(G)| \le p^{i+2}$
for all
$1 \le i \le n-4$
. Thus, by (2.1),
$d(Z(G))=1$
. Now, if
$|Z(G)|=p^3$
, then
$|Z_2(G)|=p^4$
, which contradicts (2.1). Further,
$|Z(G)|$
cannot be
$p^2$
according to [Reference Ruscitti, Legarreta and Yadav14, Theorem 4.3]. Finally, assume that
$|Z(G)|=p$
. In this case, the conclusion follows from Corollary 2.3.
The following example of a 3-group of order
$3^7$
supports Theorem 2.1.
Example 2.5. Consider the group:
$G=\langle f_1, f_2, f_3, f_4, f_5, f_6, f_7 \rangle $
with relations:
$$ \begin{align*} & f_3=[f_2,f_1], \quad f_4=f_1^3,\quad f_5=[f_3,f_1],\quad f_6=[f_3,f_2],\quad f_7=[f_5,f_1],\quad f_7^2=[f_4,f_2], \\ & f_2^3=f_3^3=f_4^3=f_5^3=f_6^3=f_7^3=[f_4,f_1]=[f_6,f_1]=[f_7,f_1]=[f_5,f_2]=[f_6,f_2] \\ &\quad =[f_7,f_2]=[f_4,f_3]=[f_5,f_3]=[f_6,f_3]=[f_7,f_3]=[f_5,f_4]=[f_6,f_4]=[f_7,f_4] \\ &\quad =[f_6,f_5]=[f_7,f_5]=[f_7,f_6]=1. \end{align*} $$
Then:
-
•
$|G|=3^7$
; -
• the nilpotency class of G is 4;
-
•
$Z(G)=\langle f_6,f_7 \rangle $
; -
•
$\Phi (G)=\langle f_3, f_4, f_5, f_6, f_7 \rangle $
; -
•
$\gamma _3(G)=\langle f_5, f_6, f_7 \rangle $
; -
•
$\gamma _4(G)=\langle f_7 \rangle $
.
Let
$x=f_5$
and
$M=C_G(x)$
. Then,
$x \in \gamma _3(G) \setminus Z(G)$
is of order 3 and
$Z(M)=\langle f_5, f_6,f_7 \rangle =\gamma _3(G)$
. Consider the automorphism defined by
$$ \begin{align*} \alpha(f_1f_4^2f_5f_6^2f_7^2) & = f_1f_4^2f_5^2f_6^2f_7^2, \\ \alpha(f_1f_3f_5f_6^2) & = f_1f_3f_5^2f_6^2, \\ \alpha(f_1^2f_2^2f_3f_4^2f_6) & =f_1^2f_2^2f_3f_4^2f_5^2f_6f_7. \end{align*} $$
By using the relators of G, we find
$\alpha (f_i)=f_i ~\text {for all}~ i \in \{ 2,3,4,5,6,7 \} ~\text {and}~ \alpha (f_1)=f_1f_5$
. It is easy to verify that
$\alpha $
is a noninner automorphism of order 3 which fixes
$\Phi (G)$
element-wise.
We conclude the paper by giving an elementary and short proof of [Reference Ghoraishi8, Theorem 1.1].
Theorem 2.6. Let G be a finite nonabelian p-group. If G fails to fulfil the condition
$Z_2^{\star }(G) \le C_G(Z_2^{\star }(G)) = \Phi (G)$
, where
$Z_2^{\star }(G)/Z(G)=\Omega _1(Z_2(G)/Z(G))$
, then G has a noninner automorphism of order p leaving
$\Phi (G)$
element-wise fixed.
Proof. Let G be a finite nonabelian p-group such that at least one of the following holds:
$Z_2^{\star }(G) \nleq C_G(Z_2^{\star }(G))$
or
$C_G(Z_2^{\star }(G)) \neq \Phi (G)$
. Assume that G does not possess any noninner automorphism of order p that fixes
$\Phi (G)$
element-wise. Observe that
$Z_2^{\star }(G)=\{ z \in Z_2(G) ~|~ z^p \in Z(G) \}$
. Let
$x \in Z_2^{\star }(G)$
,
$y \in G'$
and
$z^p \in G^p$
. Then,
$[x,y]=1$
and
$[x,z] \in Z(G)$
. Now,
$$ \begin{align*} [x,yz^p] = [x,z^p][x,y][x,y,z^p] = [x,z^p] = [x^p,z] = 1. \end{align*} $$
Therefore, it follows from [Reference Deaconescu and Silberberg5, Theorem] that
$Z_2^{\star }(G) \le C_G(\Phi (G))=Z(\Phi (G))$
. This implies that
$Z_2^{\star }(G) \le C_G(Z_2^{\star }(G))$
.
Next, we prove that
$C_G(Z_2^{\star }(G))=\Phi (G)$
. Let M be a maximal subgroup of G and let
$g \in G-M$
. Let
$z \in Z(G) \cap M$
be of order p. Then, the map
$\alpha : G \rightarrow G$
, defined by
$\alpha (mg^i)= mg^iz^i$
for all
$m \in M$
, is easily seen to be an automorphism of order p that fixes
$\Phi (G)$
element-wise. By assumption,
$\alpha =\theta _{a_M}$
, the inner automorphism induced by some
$a_M\in G$
. It is easy to check that
$a_M \in Z_2^{\star }(G)$
and
$M=C_G(a_M)$
. Since
$[Z_2^{\star }(G),\Phi (G)]=1$
, we have
$\Phi (G)\leq C_G(Z_2^{\star }(G))$
. It follows that
$$ \begin{align*} \Phi(G) \leq C_G(Z_2^{\star}(G)) \leq \bigcap \limits_{M} C_G(a_M) = \bigcap \limits_{M} M=\Phi(G). \end{align*} $$
Hence,
$Z_2^{\star }(G) \le C_G(Z_2^{\star }(G)) = \Phi (G)$
, which is a contradiction.
Acknowledgements
The first author is thankful to Dr. MSG for guidance. The second author expresses deep thanks to the National Institute of Science Education and Research, Bhubaneswar, India for supporting the post-doctoral research.
