The automorphism group of a nonsplit metacyclic 2-group

TL;DR

This paper fully characterizes the automorphism groups of nonsplit metacyclic 2-groups, filling a gap in the understanding of automorphism groups for metacyclic p-groups.

Contribution

It provides a complete description of automorphism groups for a class of nonsplit metacyclic 2-groups, advancing the classification of automorphisms in these groups.

Findings

Determined the structure of automorphism groups for nonsplit metacyclic 2-groups.

Completed the classification of automorphism groups for all metacyclic p-groups.

Extended previous work to include nonsplit cases.

Abstract

We determine the structure of automorphism group or each nonsplit metacyclic 2-group. This completes the work on automorphism groups of metacyclic $p$-groups.

Equations

$$H(n,m;t,r)=⟨\alpha ,\beta \mid {\alpha }^n=1,{\beta }^m={\alpha }^t,\beta \alpha {\beta }^{-1}={\alpha }^r⟩.$$

$$d>1\text{and}\max \{d,a-d-1\}<c<\min \{a,b\},$$

$$\max \{1,a-b\}<e<\min \{a,b\},$$

$$({\alpha }^{u_1}{\beta }^{v_1})({\alpha }^{u_2}{\beta }^{v_2})$$

$$({\alpha }^u{\beta }^v{)}^k$$

$$[x;s]\equiv (1+2^{\ell -1})x(\mathrm{mod}{2}^{\ell +u}),s^x-1\equiv (s-1+2^{2\ell -1})x(\mathrm{mod}{2}^{2\ell +u}).$$

$$2^{b-d}\mid y_1$$

$$x_2[2^b;r^{y_2}]+2^c{y}_2-x_1[2^c;r^{y_1}]-\frac{2^c{y}_1}{2^b}{2}^c\equiv 0,$$

$$(r^{y_1}-1)x_2+([r;r^{y_1}]-r^{y_2})x_1+\frac{(r-1)y_1}{2^b}{2}^c\equiv 0,$$

$$2\nmid x_1{y}_2-x_2{y}_1.$$

$$x_1$$

$$y_2$$

$$2^b{x}_2+2^c(y_2-x_1-2^{c-d}y)$$

$$2^b{x}_{2}y+(r-r^{y_2})x_1+2^{c}y$$

$$d+\Vert y_2-1\Vert =\Vert r^{y_2-1}-1\Vert \ge c\ge a-d;$$

$$r^{y_2}-r=(1+2^d)(r^{y_2-1}-1)\equiv (2^d+2^{2d-1})(y_2-1).$$

$$x_1(y_2-1)\equiv (2^{b-d}{x}_2+2^{c-d}-2^{c-1})y(\mathrm{mod}{2}^{a-d}).$$

$$(x_1+2^{c-d}y)(y_2-1-2^{c-d}y)$$

$$\equiv (2^{b-d}{x}_2-2^{c-1}+2^{c-d}(y_2-x_1-2^{c-d}y))y(\mathrm{mod}{2}^{a-d})$$

$$\equiv -2^{c-1}y(\mathrm{mod}{2}^{a-d}),$$

$$[2z;r]=(1+r)[z;r^2]\equiv 2^e(1+2^e)z\equiv 2^{e}z$$

$$r^{2z}-1=(r^2{)}^z-1\equiv (r^2-1+2^{2e+1})z$$

$$[2^b;r^{y_2}]\equiv 2^{e+b-1}\equiv 0,r^{y_1}-1\equiv 0.$$

$$e+\Vert y_2-1\Vert \ge a\text{if}2\mid y,e+\Vert y_2-1\Vert =a-1\text{if}2\nmid y;$$

$$r^{2^{b-d}y}=(1+2^d{)}^{2^{b-d}y}\equiv 1+2^{b}y.$$

$${\sigma }_{x_1,x_2;y,y_2}:{\alpha }^u{\beta }^v\mapsto {\exp }_{\alpha }(x_1[u;1+2^{b}y]+(1+2^{b}yu)x_2[v;r^{y_2}]){\exp }_{\beta }(2^{b-d}yu+y_{2}v)$$

$$y^{\prime }\equiv y(\mathrm{mod}{2}^d),x_1^{\prime }-x_1+2^{c-d}(y^{\prime }-y)\equiv 0,$$

$$y_2^{\prime }\equiv y_2(\mathrm{mod}{2}^b),x_2^{\prime }-x_2+\frac{y_2^{\prime }-y_2}{2^b}{2}^c\equiv 0.$$

$${\varphi }_{x_1,x_2}={\sigma }_{x_1,x_2;0,1},{\psi }_{y,y_2}={\sigma }_{1,0;y,y_2}.$$

$${\varphi }_{x_1^{\prime },x_2^{\prime }}\circ {\varphi }_{x_1,x_2}$$

$${\psi }_{y^{\prime },y_2^{\prime }}\circ {\psi }_{y,y_2}$$

$${\varphi }_{x_1,x_2}\circ {\psi }_{y,y_2}$$

$${\psi }_{y,y_2}\circ {\varphi }_{x_1,x_2}$$

$$X=\{{\varphi }_{x_1,x_2}:(x_1,x_2)\in \Xi \},Y=\{{\psi }_{y_1,y_2}:(y,y_2)\in \Omega \}.$$

$${\sigma }_{x_1,x_2;y,y_2}={\varphi }_{{\tilde{x}}_1,{\tilde{x}}_2}\circ {\psi }_{y,y_2},$$

$${\tilde{x}}_1=x_1-x_2[2^{b-d}y;r][y_2;r{]}^{-1},{\tilde{x}}_2=x_2[y_2;r{]}^{-1}.$$

$$\Vert y\Vert \ge d,\Vert y_2-1\Vert \ge b,x_1\equiv 1+2^{c-d}y,x_2\equiv \frac{y_2-1}{2^b}{2}^c.$$

Full text

Keywords: automorphism group, nonsplit, metacyclic 2-group.

MSC 2010: 20D45.

1 Introduction

A metacyclic group is an extension of a cyclic group by another cyclic group. As is well-known (see Section 3.7 of [8]), each metacyclic group can be presented as

[TABLE]

for some positive integers $n,m,r,t$ with $r^{m}-1\equiv t(r-1)\equiv 0\pmod{n}$.

In recent years, people have been studying automorphism groups of metacyclic $p$-groups. The order and the structure of ${\rm Aut}(G)$ when $G$ is a split metacyclic $p$-group were found in [1, 4]; when $G$ is a nonsplit metacyclic $p$-group with $p\neq 2$, ${\rm Aut}(G)$ was determined in [5].

In this paper, we deal with nonsplit metacyclic 2-groups, based on [3] in which we derived explicit formulas for automorphisms of a general metacyclic group. According to Theorem 3.2 of [7], each nonsplit metacyclic 2-group is isomorphic to one of the following:

1. (I)

$H_{I}(a,b,c,d):=H(2^{a},2^{b};2^{c},2^{d}+1)$ for a unique quadruple $(a,b,c,d)$ with

[TABLE] 2. (II)

$H_{II}(a,b,e):=H(2^{a},2^{b};2^{a-1},2^{e}-1)$ for a unique triple $(a,b,e)$ with

[TABLE] 3. (III)

$H_{III}(a):=H(2^{a},2;2^{a-1},2^{a}-1)$ for a unique $a>1$.

We do not deal with case (III), in which $G$ is a generalized quaternion group, and the automorphism group was determined in [9]. Denote $r=2^{d}+1$ in case (I), and $r=2^{e}-1,c=a-1,d=1$ in case (II). Denote $H(2^{a},2^{b};2^{c},r)$ simply as $H$ whenever there is no ambiguity.

Notation 1.1**.**

For an integer $N>0$, let $\mathbb{Z}_{N}$ denote $\mathbb{Z}/N\mathbb{Z}$, and regard it as a quotient ring of $\mathbb{Z}$. Let $\mathbb{Z}_{N}^{\times}$ denote the multiplicative group of units. For $u\in\mathbb{Z}$, denote its image under the quotient $\mathbb{Z}\twoheadrightarrow\mathbb{Z}_{N}$ also by $u$.

For integers $u,s>0$, let $[u;s]=1+s+\cdots+s^{u-1}$ and $[0;s]=0$.

For any $k,v\in\mathbb{Z}$, by $r^{k}$ we mean $r^{\check{k}}$, where $\check{k}\in\{0,1,\ldots,2^{b}-1\}$ denotes the remainder when dividing $k$ by $2^{b}$; by $[v;r^{k}]$ we mean $[\check{v};r^{\check{k}}]$. For $w\in\mathbb{Z}_{2^{a}}$, by $\alpha^{w}$ we mean $\alpha^{\tilde{w}}$ where $\tilde{w}$ is any integer mapped to $w$ by $\mathbb{Z}\twoheadrightarrow\mathbb{Z}_{2^{a}}$; similarly, $\beta^{w}$ for $w\in\mathbb{Z}_{2^{b}}$ makes sense.

For an integer $u\neq 0$, let $\|u\|$ denote the largest integer $s$ with $2^{s}\mid u$; set $\|0\|=+\infty$.

Use $\exp_{\alpha}(u)$ to denote $\alpha^{u}$ when the expression for $u$ is too long.

Throughout the paper, we abbreviate $A\equiv B\pmod{2^{a}}$ to $A\equiv B$ as often as possible.

2 Preparation

It follows from (1) that

[TABLE]

For the following two lemmas, see [3] Lemma 2.1–2.4.

Lemma 2.1**.**

If $s>1$ with $\|s-1\|=\ell>1$ and $x>0$ with $\|x\|=u>0$, then

[TABLE]

Lemma 2.2**.**

There exists $\sigma\in{\rm Aut}(H)$ with $\sigma(\alpha)=\alpha^{x_{1}}\beta^{y_{1}}$, $\sigma(\beta)=\alpha^{x_{2}}\beta^{y_{2}}$ if and only if

[TABLE]

Due to (6), we may write $y_{1}=2^{b-d}y$. The following two lemmas are special cases of [3] Lemma 2.8 (iii) and (ii), respectively. Here we reprove them in a relatively succinct way.

Lemma 2.3**.**

When $H=H_{I}(a,b,c,d)$, the conditions (7) and (8) hold if and only if

[TABLE]

Proof.

Now that $\|y_{1}\|\geq b-d>c-d\geq a-2d$, applying Lemma 2.1, we obtain $r^{y_{1}}\equiv 1+2^{b}y$, $[r]_{r^{y_{1}}}\equiv r$, $[2^{c}]_{r^{y_{1}}}\equiv 2^{c}$, $[2^{b}]_{r^{y_{2}}}\equiv 2^{b}$, hence (7) and (8) become

[TABLE]

respectively. It follows from (13) that

[TABLE]

by Lemma 2.1, $r^{y_{2}-1}-1\equiv 2^{d}(1+2^{d-1})(y_{2}-1)$, hence

[TABLE]

So (13) can be converted into $(1+2^{d-1})(y_{2}-1)x_{1}\equiv(2^{b-d}x_{2}+2^{c-d})y\pmod{2^{a-d}};$ multiplying $1-2^{d-1}$ and using (14), we obtain

[TABLE]

Then

[TABLE]

where in the last line, (12) is used. Thanks to $2\nmid x_{1}+2^{c-d}y$ and $c\geq a-d$, we can deduce (11), then (10) follows from (11), (12) and the condition $c-1\geq d\geq a-c$.

Conversely, it can be verified that (10), (11) indeed imply (12), (13). ∎

Lemma 2.4**.**

When $H=H_{II}(a,b,e)$, the conditions (7) and (8) hold if and only if one of the following occurs:

• •

$2\mid y$* and $y_{2}\equiv 1\pmod{2^{a-e}}$;*

• •

$e\leq a-2$, $2\nmid y$ and $\|y_{2}-1\|=a-e-1$.

Proof.

For any $z$ with $\|z\|=u\geq 1$, applying Lemma 2.1 to $s=r^{2}$, $x=z$, we obtain

[TABLE]

these are still true when $r$ is replaced by $r^{y_{2}}$, as $2\nmid y_{2}$ so that $\|r^{y_{2}}\pm 1\|=\|r\pm 1\|$. In particular,

[TABLE]

Note that $2^{2c-b}y_{1}\equiv 2^{c}(y_{2}-x_{1})\equiv 0$, hence (7) holds for free. The condition (8) becomes $r^{y_{2}-1}-1\equiv 2^{a-1}y$, which is, by (15), equivalent to

[TABLE]

by (9), the second possibility occurs only when $e\leq a-2$. ∎

3 The structure of automorphism group

3.1 $H=H_{I}(a,b,c,d)$

Let $\Xi$ denote the set of $(x_{1},x_{2})\in\mathbb{Z}_{2^{a}}\times\mathbb{Z}_{2^{a}}$ satisfying (10), and let $\Omega$ denote the set of $(y,y_{2})\in\mathbb{Z}_{2^{a+d-c}}\times\mathbb{Z}_{2^{a+b-c}}$ satisfying (11). By Lemma 2.1 and (2), we have

[TABLE]

Thus each automorphism of $H_{I}(a,b,c,d)$ can be expressed as

[TABLE]

for some quadruple $(x_{1},x_{2},y,y_{2})\in\Xi\times\Omega$, and $\sigma_{x_{1},x_{2};y,y_{2}}=\sigma_{x^{\prime}_{1},x^{\prime}_{2};y^{\prime},y^{\prime}_{2}}$ if and only if

[TABLE]

Let

[TABLE]

The following can be verified using (4), (5):

[TABLE]

Let

[TABLE]

Then $X,Y$ are subgroups of ${\rm Aut}(H)$, and there is a decomposition

[TABLE]

As special cases of (18) and (19), $\phi_{x_{1},x_{2}}=\psi_{y,y_{2}}$ if and only if

[TABLE]

It follows that

[TABLE]

Let

[TABLE]

Consider the homomorphism $\rho:X\to\mathbb{Z}_{2^{a}}$ sending $\phi_{x_{1},x_{2}}$ to $x_{1}$. It is easy to see that

[TABLE]

If $f\geq c+2$, then ${\rm Im}(\rho)=\langle 1-2^{f-c}\rangle\cong\mathbb{Z}_{2^{a+c-f}}$, hence

[TABLE]

if $f=c+1$, then ${\rm Im}(\rho)=\mathbb{Z}_{2^{a}}^{\times}=\langle-1,5\rangle\cong\mathbb{Z}_{2}\times\mathbb{Z}_{2^{a-2}}$ by Theorem 2’ on Page 43 of [6]. Hence

[TABLE]

Consider the homomorphism $\mu:Y\to\mathbb{Z}_{2^{b+a-c}}$ sending $\psi_{y_{1},y_{2}}$ to $y_{2}$. Clearly

[TABLE]

If $c\geq d+2$, then ${\rm Im}(\mu)=\langle 1-2^{c-d}\rangle\cong\mathbb{Z}_{2^{a+b+d-2c}}$, hence

[TABLE]

if $c=d+1$, then ${\rm Im}(\mu)=\mathbb{Z}_{2^{b+a-c}}^{\times}=\langle-1,5\rangle\cong\mathbb{Z}_{2}\times\mathbb{Z}_{2^{a+b-c-2}}$, hence

[TABLE]

The above can be summarized as

Theorem 3.1**.**

For $H=H_{I}(a,b,c,d)$, we have ${\rm Aut}(H)=XY$ such that

[TABLE]

Consequently, the order of ${\rm Aut}(H)$ is $2^{b+c+2d}$.

Remark 3.2**.**

In principle, we are able to obtain a presentation for ${\rm Aut}(H)$. The generators are those of $X,Y$, and the relator set can be divided into $R_{X}\cup R_{Y}\cup R_{X,Y}\cup R^{\prime}$, where

• •

$R_{X}$ (resp. $R_{Y}$) consists of relators in the presentations for $X$ (resp. $Y$),

• •

$R_{X,Y}$ consists of two elements corresponding to the two generators of $X\cap Y$, i.e., $\phi_{1+2^{c},0}=\psi_{2^{d},1}$ and $\phi_{1,2^{c}}=\psi_{0,1+2^{b}}$,

• •

relators in $R^{\prime}$ have the form $\vartheta_{j}\theta_{i}=\gamma\delta$, where $\vartheta_{j}$ (resp. $\theta_{i}$) is a generator for $Y$ (resp. $X$), and $\gamma$ (resp. $\delta$) is a product of generators for $X$ (resp. $Y$).

As an example of element in $R^{\prime}$,

[TABLE]

one can further write $\phi_{h^{-1},2^{a-f}h^{-1}}$ (resp. $\psi_{2^{a-c},h}$) as a product of generators for $X$ (resp. $Y$).

However, the computations are so complicated that we choose not to write down explicitly.

3.2 $H=H_{II}(a,b,e)$

Recall that $r=2^{e}-1$, $c=a-1$ and $d=1$.

Let $\Xi$ denote the set of $(x_{1},x_{2})\in\mathbb{Z}_{2^{a}}\times\mathbb{Z}_{2^{a}}$ with $2\nmid x_{1}$, and let $\Omega$ denote the set of $(y,y_{2})\in\mathbb{Z}_{4}\times\mathbb{Z}_{2^{b+1}}$ satisfying the conditions in Lemma 2.4. Each automorphism of $H_{II}(a,b,e)$ can be expressed as (recalling $r^{y_{1}}\equiv 1$, as in (16))

[TABLE]

for some quadruple $(x_{1},x_{2},y,y_{2})\in\Xi\times\Omega$, and $\sigma_{x_{1},x_{2};y,y_{2}}=\sigma_{x^{\prime}_{1},x^{\prime}_{2};y^{\prime},y^{\prime}_{2}}$ if and only if

[TABLE]

Let

[TABLE]

We have

[TABLE]

Let

[TABLE]

Thus $X,Y$ are subgroups of ${\rm Aut}(H)$, and there is a decomposition

[TABLE]

As special cases of (43), (44), $\phi_{x_{1},x_{2}}=\psi_{y,y_{2}}$ if and only if

[TABLE]

Hence

[TABLE]

Let

[TABLE]

Similarly as in the previous subsection, we can obtain

[TABLE]

If $e\leq a-3$, then

[TABLE]

If $e=a-2$, then

[TABLE]

In the notations of [2] Chapter IV, Section 3, choose a map $s:\mathbb{Z}_{2}\times\mathbb{Z}_{2^{b-1}}\to Y$, $(u,v)\mapsto\psi_{1}^{u}\psi_{2}^{v}$, so that $s(0,0)=1$, then it can be computed that

[TABLE]

Hence $Y$ is isomorphic to the central extension of $\mathbb{Z}_{2}\times\mathbb{Z}_{2^{b-1}}$ by $\mathbb{Z}_{2}$ determined by the 2-cocycle

[TABLE]

denote this group by $E_{f}$.

If $e=a-1$, then we find directly that

[TABLE]

Theorem 3.3**.**

For $H=H_{II}(a,b,e)$, we have ${\rm Aut}(H)=XY$ such that

[TABLE]

Consequently, the order of ${\rm Aut}(H)$ is $2^{a+b+\min\{a-2,e\}}$.