ON NONINNER AUTOMORPHISMS OF SOME FINITE P-GROUPS

Abstract

We settle the noninner automorphism conjecture for finite p-groups ($p> 2$) with certain conditions. Also, we give an elementary and short proof of the main result of Ghoraishi [‘On noninner automorphisms of finite nonabelian p-groups’, Bull. Aust. Math. Soc. 89(2) (2014) 202–209].

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 [11, 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 [12]. 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 [1] in 2007. In 2013, Abdollahi et al. [3, 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 [15] 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. [4] 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 [5–10].

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 [13, Lemma 9.108]). In 2002, Deaconescu and Silberberg [5] 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 [5, 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. [14] 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 [8, Theorem 1.1], Ghoraishi improved the reduction given by Deaconescu and Silberberg [5] 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 [2, Corollary 2.3] that

(2.1) $$ \begin{align} d(Z_2(G)/Z(G))=d(Z(G))\,d(G). \end{align} $$

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 [14, 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 [8, 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 [5, 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.