The Amit–Ashurst conjecture for finite metacyclic p-groups

Abstract

The Amit conjecture about word maps on finite nilpotent groups has been shown to hold for certain classes of groups. The generalised Amit conjecture says that the probability of an element occurring in the image of a word map on a finite nilpotent group G is either 0, or at least 1/|G|. Noting the work of Ashurst, we name the generalised Amit conjecture the Amit–Ashurst conjecture and show that the Amit–Ashurst conjecture holds for finite p-groups with a cyclic maximal subgroup.

1 Introduction

Recall that a word w in k variables is an element of some free group ${{\textbf {F}}}_k= \langle x_1,\dots ,x_k\rangle$. The word w induces a word map on a finite group G, which by abuse of notation we write as

where $G^k$ denotes the direct product of k copies of G. This map induces a probability distribution $P_{w,G}$ on G, where for $g\in G$,

We also write

$$\begin{aligned} N_{w,G}(g)=\bigl |\bigl \{(g_1,\dots ,g_k)\in G^k \,{|}\, w(g_1,\ldots ,g_k)=g\bigr \}\bigr |, \end{aligned}$$

or simply $N_w(g)$ when the group is clear.

We denote by $G_w$ the set of word values of w in G, i.e. the set of elements $g\in G$ such that the equation $w=g$ has a solution in $G^k$. Additionally, the fibre of g in $G^k$ is

$$\begin{aligned} \bigl \{(g_1,\ldots , g_k)\in G^k\,{|}\, w(g_1,\ldots , g_k)=g\bigr \}. \end{aligned}$$

Nikolov and Segal [11, Theorem 1.3] gave a characterisation of finite nilpotent groups (and of finite solvable groups) based on the probability function $P_{w,G}$: A finite group G is nilpotent if and only if the values $P_{w,G}(g)$ are bounded away from zero as g ranges over $G_w$ and w ranges over all group words. The bound is quite small; of the order $p^{-|G|}$ where p is the largest prime divisor of |G|.

This bound has been improved in some cases. For example, it is easy to show that for an abelian group G, a word w and g in $G_w$, then $P_{w,G}(g)\geqslant |G|^{-1}$, since any word map is a homomorphism over an abelian group. Additionally, Ashurst [1, Chapter 4] showed that the nilpotent dihedral groups and the generalised quaternion groups also satisfy this property. In this direction, Ashurst conjectured the following in her thesis [1, Conjecture 6.2.1], which we call the Amit–Ashurst conjecture:

Conjecture

(Amit–Ashurst) Let G be a finite nilpotent group. Then

$$\begin{aligned} P_{w,G}(g) \geqslant |G|^{-1}, \end{aligned}$$

where w ranges over all words and $g\in G_w$.

The Amit–Ashurst conjecture is a generalisation of the Amit conjecture which asks whether for a finite nilpotent group G, the probability distribution induced by the word w satisfies $P_{w,G}(1)\geqslant |G|^{-1}$ for all w. We note that for all groups G, and all $g \in G$, the word $w = x$ satisfies $P_{w,G}(g)=|G|^{-1}$. Hence, the bound proposed in the Amit–Ashurst conjecture is strict. The Amit–Ashurst conjecture first appeared as the generalised Amit conjecture in [3]. However, since Ashurst first asked the question, we believe the term Amit–Ashurst conjecture is more appropriate.

Not much progress has been made on the Amit conjecture, or the Amit–Ashurst conjecture. Since a finite nilpotent group is a direct product of its Sylow subgroups, we note that it suffices to consider finite p-groups for either of these conjectures. It is known that class 2 groups satisfy the Amit conjecture [6, 9] as do split metacyclic groups [9, Proposition 1.3]. However, besides the examples given by Ashurst, there were no other families of groups known to satisfy the Amit–Ashurst conjecture. In this paper we show that the Amit–Ashurst conjecture holds for finite p-groups with a cyclic maximal subgroup. Formally our theorem is:

Theorem A

For p a prime, let G be a finite p-group with a cyclic maximal subgroup. Then for any word w and $g\in G_w$ we have that $P_{w,G}(g) \geqslant |G|^{-1}$.

Our proof of Theorem A is broken into two parts. First we show that word maps with large images have well-behaved probability distributions on a larger family of finite metacyclic p-groups. Then, recalling the classification of finite p-groups with a cyclic maximal subgroup, we show that word maps with minimal nontrivial image have appropriately behaved probability distributions.

Notation

Throughout, we use left-normed commutators, for example, $[x,y,z] = [[x,y],z]$. Furthermore we define $[x,y]=xyx^{-1}y^{-1}$.

2 Word maps over metacyclic groups

Our basic set up is $G = AB$ is a finite p-group, where with A and B both cyclic. We write Z for the unique subgroup of order p in A. As suggested by the notation, the subgroup Z is central.

As often is the case for nilpotent groups, we would like to prove the Amit–Ashurst conjecture by taking the quotient by Z (or any other normal subgroup) and looking at G/Z. Assuming by induction that $P_{w,G/Z}(gZ)$ is at least $|G/Z|^{-1}$, then one would hope that the probability distributes over the coset gZ nicely. This does not work in general, but is an illustrative technique.

First we document a general instance where this idea works.

Lemma 2.1

Let G be a group and N a normal subgroup such that G/N satisfies the Amit–Ashurst conjecture. Suppose w is a word in k variables and $g \in G_w$ is such that $N_w(gn) = N_w(g)$ for all $n \in N$. Then $N_w(g) \geqslant |G|^{k-1}$.

Proof

Let w be a word in k variables. Write ${\overline{G}} = G/N$ and we will use the bar notation for elements in ${\overline{G}}$. A simple counting argument, see [1, p. 20], shows that

By assumption $N_{w, G/N}({\bar{g}}) \geqslant |G/N|^{k-1}$. Thus

and the result follows. $\square$

Lemma 2.2

Let G be a finite p-group, such that $G = AB$, where and both A and B are cyclic. Let w be a word. If for all $b\in B$, then one of the following holds:

• (1) $G_w = G$,

• (2) $G_w\subseteq A$.

Proof

By Hall’s collection process [7, Section 11.1], note that for some $\ell \in {\mathbb {N}}\cup \{0\}$, $e_1,\ldots ,e_\ell \in {\mathbb {Z}}$ and word $\kappa$ that is a product of commutators. Since G/A is cyclic, we see that . Hence the image of $\kappa$ is contained in A. Let $d=\gcd {(e_1,\dots ,e_\ell )}$. Since , if $p\,{|}\, d$ then the image is entirely contained in A; otherwise, we have $(p,d)= 1$ and w is a surjective map. $\square$

In particular, if $G_w = G$ then the Amit–Ashurst conjecture follows from [4, Lemma 7]. Therefore, for the rest of this section we consider the case when $G_w \subset G$.

Let $G = AB$ be a finite p-group, where with A and B both cyclic. For $\mathbf{{b}} = (b_1, \ldots , b_k)\in B^k$, we consider the coset ${\textbf{b}}A^k$ in $G^k$. Recall that A is normal in G and as $A=\langle a\rangle$ is cyclic, the cyclic subgroup $B=\langle b\rangle$ acts on A by powers, i.e. for some $\ell \in {\mathbb {Z}}$. Hence over the coset ${\textbf{b}}A^k$, a word w acts as

(2.1)

where $v_{w,\mathbf{{b}}}$ is a word map over A and $w_{\mathbf{{b}}}$ is $v_{w,\mathbf{{b}}}$ times $b^i$ for some i.

Lemma 2.3

Let G be a finite p-group, such that $G = AB$, where and both A and B are cyclic. Let Z be the unique subgroup of order p in A. Suppose that $A\cap B \leqslant Z$ and that G/Z satisfies the Amit–Ashurst conjecture. Then any word w such that

$$\begin{aligned} G_w \not \subseteq Z\quad \text {and}\quad G_w \subseteq A, \end{aligned}$$

has $Z \subsetneq G_w$ and satisfies $P_{w,G}(g) \geqslant |G|^{-1}$ for all $g \in G_w$.

Proof

For $\mathbf{{b}} = (b_1, \ldots , b_k)\in B^k$, we have from (2.1) that

As $G_w\subseteq A$, we observe that $w({\textbf{b}})\in Z$. Also, as A is abelian is a homomorphism with image, $A_{v_{w,\textbf{b}}}$, a subgroup of A and all fibres of $v_{w,\textbf{b}}$ are of the same size. Note $A_{v_{w,\textbf{b}}}$ is either trivial or contains Z, in the second case $A_{v_{w,\textbf{b}}}$ is a union of cosets of Z. Thus for any $g \in G_w \backslash Z$ and any fixed $\mathbf{{b}}$ it follows that $N_{ v_{w,\textbf{b}}}(g) = N_{ v_{w,\textbf{b}}}(gz) \leqslant N_{ v_{w,\textbf{b}}}(z)$ for all $z \in Z$, with the inequality coming from the case when $g \not \in A_{v_{w,\textbf{b}}}w(\mathbf{{b}})$ but $z \in A_{v_{w,\textbf{b}}}w(\mathbf{{b}})$. Summing over the $\textbf{b}$, gives $N_w(g) = N_w(gz) \leqslant N_w(z)$.

We can now proceed by induction. Let ${\overline{G}} = G/Z$ (note Z is central in G). Suppose $g \in G_w \backslash Z$. By hypothesis $N_w({\bar{g}}) \geqslant |{\overline{G}}|^{k-1}$. By the previous paragraph $N_w(g) = N_w(gz)$ for all $z \in Z$ and by Lemma 2.1 it follows that $N_{w}(g) \geqslant |G|^{k-1}$. Furthermore, as $N_w(z) \geqslant N_w(g)$ for all $z \in Z$, both the results follow. $\square$

3 The polynomial argument

The previous section was largely inspired by Levy [9]. The impetus for this section originates in Ashurst’s dissertation and together with Lemma 2.3 can be used to prove the Amit–Ashurst conjecture for certain metacyclic p-groups [1]. Recall that Lemma 2.3 completes the idealised inductive proof of the Amit–Ashurst conjecture for finite metacyclic p-groups, except for the case when $1\ne G_w\subseteq Z$. To handle this case we consider the following argument.

Let G be a finite nonabelian metacyclic p-group. Then from [8] the group G can be represented as

where $r =p^{n - \delta } + 1$ or $r = p^{n - \delta } -1$ (called positive and negative type respectively) for $n,m\in {\mathbb {N}}$ and $\epsilon ,\delta \in \{0,1,\ldots ,n\}$ such that $p^n$ divides both ${p^{n-\epsilon }}(r-1)$ and . Also, if , note using $b^{\beta }a^{\alpha } = a^{\alpha r^{\beta }}b^{\beta }$ for $0\leqslant \alpha <p^n$ and $0 \leqslant \beta < p^m$, we get

(3.1)

which will be of later use.

Suppose that a word w maps G to $Z = \langle a^{p^{n-1}} \rangle$, the unique central subgroup of order p in $\langle a \rangle$. So we have

for some function where $\varvec{\alpha } = (\alpha _1, \ldots , \alpha _k)$, $\varvec{\beta }= (\beta _1, \ldots , \beta _k)$ with $0 \leqslant \alpha _i < p^{n}$ and $0 \leqslant \beta _i < p^m$.

Suppose for now that we can replace f with a polynomial .

We note the following useful result.

Theorem 3.1

(Chevalley–Warning, [10, Theorem 6.11]) Let p be a prime, let $\ell \in {\mathbb {N}}$ and let ${\widetilde{q}}\in {\mathbb {F}}_p[t_1,\ldots ,t_\ell ]$ be a polynomial with . If the number N of $(\alpha _1,\ldots ,\alpha _\ell )\in {\mathbb {F}}_p^\ell$ with ${\widetilde{q}}(\alpha _1, \ldots ,\alpha _\ell )=0$ satisfies $N\geqslant 1$, then $N\geqslant p^{\ell -d}$.

For the purposes of the Amit–Ashurst conjecture, we aim to apply the Chevalley–Warning Theorem to some of the polynomials

$$\begin{aligned} {\widetilde{q}}_i(t_1,\ldots ,t_{2k})=q(t_1,\ldots ,t_{2k})-i\quad \text {for each}\;\; i\in {\mathbb {F}}_p; \end{aligned}$$

in this setting, the number of variables is $\ell =2k$. In order to do so, we will first need that . Either this holds, or if not we can replace w with ${\widehat{w}}(x_1, \ldots , x_{\mu }) = w(x_1, \ldots , x_k)$ where $\mu > k$. We note that, as a word map, the word ${\widehat{w}}$ sends $(g_1,\ldots ,g_{\mu })$ to $w(g_1,\ldots ,g_k)$. Hence ${\widehat{w}}$ has the same image as w and also each element in the image has the same probability of occurring since $x_{k+1}, \ldots , x_{\mu }$ can take any value in G.

Next, as we will see below, we need a sharper upper bound on . Write c for the nilpotency class of G. We suppose for now that , and as seen in the previous paragraph we may assume that $2k> c$.

To summarise, we suppose

for some polynomial , where $\overline{\varvec{\alpha }} \equiv \varvec{\alpha } \ (\textrm{mod}\, p)$ and $\overline{\varvec{\beta }} \equiv \varvec{\beta } \ (\textrm{mod}\,p)$, with . Suppose $G_w=\{(a^{p^{n-1}})^i\,{|}\, i\in I\}\subseteq Z$, for some subset $I\subseteq {\mathbb {F}}_p$. Note that for each $i\in I$, the equation $q(\overline{\varvec{\alpha }},\overline{\varvec{\beta }})=i$ has at least one solution. For each $i\in I$, we set ${\widetilde{q}}_i(\overline{\varvec{\alpha }},\overline{\varvec{\beta }})=q(\overline{\varvec{\alpha }},\overline{\varvec{\beta }})-i$. Consequently, by Theorem 3.1, for any $g\in G_w$, the set of solutions for $w=g$ (as seen as a word in k variables) is of size at least . Indeed, recalling that

for each solution $(\overline{\varvec{\alpha }},\overline{\varvec{\beta }})$ of ${\widetilde{q}}_i$, where $i\in I$ is such that $g=(a^{p^{n-1}})^i$, we have $p^{k(n-1)}$ choices for $\varvec{\alpha }$ with $\varvec{\alpha } \equiv \overline{\varvec{\alpha }} \ (\textrm{mod}\, p)$ and $p^{k(m-1)}$ choices for $\varvec{\beta }$ with $\varvec{\beta } \equiv \overline{\varvec{\beta }} \ (\textrm{mod}\, p)$. Hence,

which gives us the Amit–Ashurst bound for the case $G_w\subseteq Z$.

Therefore it remains to determine the instances when we can replace $f(\varvec{\alpha },\varvec{\beta })$ with a polynomial of degree at most c. This will be done in the next section, but we record here some key results for our goal.

Recall from Hall’s collecting process [7, Section 11.1] that

where $\kappa (x_1,\ldots ,x_k) \in {{\textbf {F}}}_k'$ and $e_1,\ldots ,e_k\in {\mathbb {Z}}$. As w maps to Z we have $p^{n-1}$ divides $e_i$, for $1\leqslant i\leqslant k$. Next, since G is metabelian we only need to consider the basic commutators $\gamma _{\ell }(g_1, \ldots , g_{\ell }) = [g_1, g_2, \ldots , g_{\ell }]$ for $\ell \geqslant 2$. Thus we now analyse the commutator structure of G, remembering that we define a commutator [g, h] as $ghg^{-1}h^{-1}$. As any element in G can be written as $a^{\alpha _i} b^{\beta _j}$ for some $0 \leqslant \alpha _i < p^n$ and $0 \leqslant \beta _j < p^m$, we introduce the following notation: for $\ell \geqslant 2$,

for $\varvec{\alpha } = (\alpha _1, \ldots , \alpha _{\ell })$ and $\varvec{\beta }= (\beta _1, \ldots , \beta _{\ell })$, where $q_\ell (\varvec{\alpha },\varvec{\beta })$ is some function in $\varvec{\alpha }$ and $\varvec{\beta }$. Note our abuse of notation where if $\varvec{\alpha } = (\alpha _1, \ldots , \alpha _t)$ and $\varvec{\beta } = (\beta _1, \ldots , \beta _t)$ and $\ell <t$ then $q_\ell (\varvec{\alpha }, \varvec{\beta }) = q_\ell ((\alpha _1, \ldots , \alpha _{\ell }), (\beta _1, \ldots , \beta _{\ell }))$.

Lemma 3.2

In the set-up above, suppose G is a metacyclic p-group with

Then

$$\begin{aligned} q_2(\varvec{\alpha }, \varvec{\beta }) = \alpha _1(1-r^{\beta _2}) - \alpha _2(1-r^{\beta _1}) \end{aligned}$$

and

$$\begin{aligned} q_{\ell }(\varvec{\alpha }, \varvec{\beta }) = q_2(\varvec{\alpha }, \varvec{\beta })(1-r^{\beta _3}) \cdots (1-r^{\beta _{\ell }}) \end{aligned}$$

for $\ell \geqslant 3$.

Proof

As $ba = a^rb$ it follows that $b^{\beta } a^{\alpha } = a^{\alpha r^{\beta }}b^{\beta }$. Thus

This gives the expression for $q_2(\varvec{\alpha }, \varvec{\beta })$. We then proceed by induction. Suppose $\ell \geqslant 3$. We have

$\square$

We can apply the lemma in various different cases.

Lemma 3.3

Suppose G is a metacyclic 2-group of one of the following types:

or

Then for $\ell \geqslant 2$,

$$\begin{aligned} q_{\ell }(\varvec{\alpha }, \varvec{\beta }) = 2^{\ell -1} {\bar{\beta }}_{\ell } \cdots {\bar{\beta }}_3(\alpha _1 {\bar{\beta }}_2 - \alpha _2 {\bar{\beta }}_1) \end{aligned}$$

where ${\bar{\beta }}_i \equiv \beta _i \ (\textrm{mod}\, 2)$ and ${\bar{\beta }}_i\in {\mathbb {F}}_2$ for $1\leqslant i\leqslant \ell$.

Proof

We apply Lemma 3.2 with $r = -1$ (equivalently ). So

$$\begin{aligned} q_2(\varvec{\alpha }, \varvec{\beta }) = \alpha _1 (1 - (-1)^{\beta _2}) - \alpha _2(1 - (-1)^{\beta _1}) = 2(\alpha _1 {\bar{\beta }}_2 - \alpha _2 {\bar{\beta }}_1), \end{aligned}$$

and

$\square$

In particular the above lemma can be applied to dihedral groups (when $b^2 =1$) and generalised quaternion groups (when $b^2 = a^{2^{n-1}}$); in these cases $\bar{\beta _i} = \beta _i$.

The following lemma considers semidihedral groups.

Lemma 3.4

Let for $n\geqslant 2$. Then for $\ell \geqslant 2$,

$$\begin{aligned} q_{\ell }(\varvec{\alpha }, \varvec{\beta }) = 2^{\ell -1} (1 - 2^{n-2})^{\ell -1} \beta _3 \cdots \beta _{\ell }(\alpha _1 \beta _2 - \alpha _2 \beta _1). \end{aligned}$$

Proof

Here we apply Lemma 3.2 with , and we obtain

and for $\ell > 2$

$\square$

Note that in the above result, we have $\beta _\ell \in {\mathbb {F}}_2$ since b has order 2.

Lemma 3.5

Suppose G is a metacyclic p-group with

Then either $\epsilon =0$ or $G\cong Q_8$.

Furthermore,

$$\begin{aligned} q_2(\varvec{\alpha }, \varvec{\beta }) = p^{n-1}(\alpha _2 \beta _1 - \alpha _1 \beta _2). \end{aligned}$$

Proof

For the first claim, we observe that G is of class 2. The claim now follows from [2, Theorem 1.1]. For the second claim, we proceed as above, noting that . $\square$

Remark 3.6

For a group G as in the above lemma, when p is odd and if $m=1$ (equivalently, if G is a nonabelian p-group of order $p^{n+1}$ with an element a of order $p^{n}$, for an odd prime p), then it follows from [12, Theorem 7.18] that G is a semidirect product of $\langle a \rangle$ and $\langle b \rangle$ where $o(b) = p$ and $a^b = a^{1+p^{n-1}}$.

4 The Amit–Ashurst conjecture for finite p-groups with cyclic maximal subgroups

A nonabelian finite p-group G has a cyclic maximal subgroup if and only if there are cyclic subgroups A and B with , the product $AB = G$, and $|G\,{:}\,A| = p$. There are five such families of p-groups [5, Theorem 5.4.4].

The nonabelian 2-groups with a cyclic maximal subgroup are:

• (1) the generalised quaternion groups

  

• (2) the dihedral groups

  

• (3) the semidihedral groups

  

• (4) the groups of the form

  

For an odd prime p, the nonabelian p-groups with a cyclic maximal subgroup are:

• (5) the groups of the form

  

In addition to differences between 2 and odd primes, nonabelian finite p-groups with cyclic maximal subgroups can be broadly separated into two categories: those of class 2 and those of maximal class. The dihedral, generalised quaternions, and semidihedral groups have maximal class, while the other nonabelian finite p-groups with cyclic maximal subgroups have class 2. We observe that quotienting by the central subgroup of order p results in either a maximal class subgroup of smaller rank, or an abelian group. This allows us to use Lemma 2.3, assuming that the Amit–Ashurst conjecture holds for the quotient. We note that the Amit–Ashurst conjecture holds for all finite abelian groups.

Proposition 4.1

For p a prime, let G be a finite nonabelian p-group with a cyclic maximal subgroup. Then, for any word w such that $G_w\subseteq Z$, where Z is the minimal central subgroup of order p, we have $P_{w,G}(g) \geqslant |G|^{-1}$ for all $g \in G_w$.

Proof

Suppose w is a word in k variables such that $1\ne G_w\subseteq Z$. Then, using the argument and the notation from the previous section, it suffices to show that

for some polynomial of degree at most the nilpotency class of G, where $\overline{\varvec{\alpha }} \equiv \varvec{\alpha } \ (\textrm{mod}\, p)$ and $\overline{\varvec{\beta }} \equiv \varvec{\beta } \ (\textrm{mod}\, p)$.

Note from the above classification of finite nonabelian p-groups with a cyclic maximal subgroup, we have that either (1) G is dihedral, (2) G is generalised quaternion, (3) G is semidihedral or (4)-and-(5) . In all cases $|G| = p^{n+1}$ and we are assuming our word w maps to $\langle a^{p^{n-1}} \rangle$. Recall

It follows that $p^{n-1}$ divides $e_i$. Also using (3.1), one can check that

where $\beta _i\equiv {\overline{\beta }}_i \ (\textrm{mod}\, p)$ and ${\overline{\beta }}_i\in {\mathbb {F}}_p$. Hence

for some polynomial of degree at most 2. Recall also that the word $\kappa$ is a product of basic commutators of length at most the nilpotency class of G. Note $\kappa$ is of this form since G is metabelian. We consider

where ${\Gamma }_j(\varvec{\alpha }, \varvec{\beta })$ is the product of all basic commutators in j variables in the word $\kappa$. Also, by abuse of notation, we do not distinguish the fact that the effective set of variables for each of the maps $\rho$, $\kappa$ and all ${\Gamma }_j$ is some subset of the respective given set of variables.

If G is of type (4) or (5), only ${\Gamma }_2(\varvec{\alpha }, \varvec{\beta })$ appears and the result follows since $q_2(\varvec{\alpha }, \varvec{\beta } )$ is of the required form by Lemma 3.5. In the remaining three cases, we have that G is a 2-group. As $G_w\subseteq \langle a^{2^{n-1}} \rangle$, we deduce that ${\Gamma }_2(\varvec{\alpha }, \varvec{\beta })\in \langle a^{2^{n-1}}\rangle$ for all choices of $\varvec{\alpha }, \varvec{\beta }$. Indeed, for ${\Gamma }_2(\varvec{\alpha }, \varvec{\beta })$ to be nontrivial, we need some of $\alpha _{i_1}, \alpha _{i_2},\beta _{i_1},\beta _{i_2}$ to be nonzero, for some choice of distinct $i_1,i_2\in \{1,\ldots , k\}$; compare Lemmas 3.3 and 3.4, noting also that we may assume that $\beta _{i_1},\beta _{i_2}\in \{0,1\}$. For $1\leqslant i_1<i_2\leqslant k$, let

and we set

$$\begin{aligned} {\textbf{I}}=\bigl \{(i_1,i_2)\mid {\Lambda }_2(i_1,i_2)\ne 1 \text { for some }\alpha _{i_1}, \alpha _{i_2},\beta _{i_1},\beta _{i_2}\bigr \}. \end{aligned}$$

Then

For each fixed choice of such $i_1,i_2$ with ${\Gamma }_2(\varvec{\alpha }, \varvec{\beta })\ne 1$, set $\alpha _i=\beta _i=0$ for all $i\in \{1,\ldots , k\}\backslash \{i_1,i_2\}$. It then follows that ${\Gamma }_j(\varvec{\alpha }, \varvec{\beta })=1$ for all $j>2$. Next, suppose that ${\Gamma }_2(\varvec{\alpha }, \varvec{\beta })$ is a product of commutators of lengths $\ell _1,\ldots ,\ell _d$ for some $d\in {\mathbb {N}}$ with $2\leqslant \ell _1<\cdots <\ell _d\leqslant n$; here n is also the nilpotency class of G. For $2\leqslant \ell \leqslant n$, it follows from Lemmas 3.3 and 3.4 that $2^{\ell -1}\,{|}\, q_\ell (\varvec{\alpha }, \varvec{\beta })$ for all $\varvec{\alpha }, \varvec{\beta }$. Hence if ${\Lambda }_2(i_1,i_2)$ is nontrivial for nonzero $\alpha _{i_1},\beta _{i_2}$ with $\beta _{i_1}$ being zero, we have that

$$\begin{aligned} {\Lambda }_2(i_1,i_2)=a^{\alpha _{i_1} (2^{\ell _1-1}\mu _1+\cdots + 2^{\ell _d-1}\mu _d)} \end{aligned}$$

for some $\mu _1,\ldots ,\mu _d\in {\mathbb {Z}}$; here the $\mu _t$ correspond to the exponent of the commutator $[a^{\alpha _{i_2}}b^{\beta _{i_2}},a^{\alpha _{i_1}}b^{\beta _{i_1}}, a^{\alpha _{i_2}}b^{\beta _{i_2}},\overset{\ell _t-2}{\ldots },a^{\alpha _{i_2}}b^{\beta _{i_2}} ]$ in ${\Lambda }_2(i_1,i_2)$, equivalently in ${\Gamma }_2(\varvec{\alpha }, \varvec{\beta })$, for $t\in \{1,\ldots , d\}$. The condition $G_w\subseteq \langle a^{2^{n-1}} \rangle$ now implies that

$$\begin{aligned} 2^{\ell _1-1}\mu _1+\cdots + 2^{\ell _d-1}\mu _d\equiv 0\pmod {2^{n-1}}; \end{aligned}$$

indeed, let for example $\alpha _{i_1}=1$. If ${\Lambda }_2(i_1,i_2)$ is nontrivial for nonzero $\alpha _{i_2},\beta _{i_1}$ with $\beta _{i_2}$ being zero, then

$$\begin{aligned} {\Lambda }_2(i_1,i_2)=a^{-\alpha _{i_2} (2^{\ell _1-1}\nu _1+\cdots + 2^{\ell _d-1}\nu _d)} \end{aligned}$$

for some $\nu _1,\ldots ,\nu _d\in {\mathbb {Z}}$, here the $\nu _t$ correspond to the exponent of the commutator $[a^{\alpha _{i_1}}b^{\beta _{i_1}},a^{\alpha _{i_2}}b^{\beta _{i_2}}, a^{\alpha _{i_1}}b^{\beta _{i_1}},\overset{\ell _t-2}{\ldots },a^{\alpha _{i_1}}b^{\beta _{i_1}} ]$ in ${\Lambda }_2(i_1,i_2)$. Similarly

$$\begin{aligned} 2^{\ell _1-1}\nu _1+\cdots + 2^{\ell _d-1}\nu _d\equiv 0\pmod {2^{n-1}}. \end{aligned}$$

Finally, if ${\Lambda }_2(i_1,i_2)$ is nontrivial for nonzero $\beta _{i_1},\beta _{i_2}$, the condition $G_w\subseteq \langle a^{2^{n-1}} \rangle$, together with the fact that

$$\begin{aligned} {\Lambda }_2(i_1,i_2)=a^{(\alpha _{i_1}-\alpha _{i_2}) (2^{\ell _1-1}\lambda _1+\cdots + 2^{\ell _d-1}\lambda _d)} \end{aligned}$$

for some $\lambda _1,\ldots ,\lambda _d\in {\mathbb {Z}}$, yields that

$$\begin{aligned} 2^{\ell _1-1}\lambda _1+\cdots + 2^{\ell _d-1}\lambda _d\equiv 0\pmod {2^{n-1}}; \end{aligned}$$

here the $\lambda _t$ correspond to the total exponent of all commutators of length $\ell _t$ in ${\Lambda }_2(i_1,i_2)$. Therefore ${\Gamma }_2(\varvec{\alpha }, \varvec{\beta })= (a^{2^{n-1}})^{{\overline{q}}_2(\overline{\varvec{\alpha }}, \overline{\varvec{\beta }} )}$ for some polynomial of degree at most 3. Specifically, we have

where the $\mu _t,\nu _t,\lambda _t$ depend on $(i_1,i_2)\in {\textbf{I}}$. Repeating this argument recursively for increasing values of $j\geqslant 2$, we deduce that ${\Gamma }_j(\varvec{\alpha }, \varvec{\beta })= (a^{2^{n-1}})^{{\overline{q}}_j(\overline{\varvec{\alpha }}, \overline{\varvec{\beta }} )}$ for some polynomial of degree at most $j+1$, for all $2\leqslant j< n$. Lastly, from Lemmas 3.3 and 3.4, it is clear that ${\Gamma }_n(\varvec{\alpha }, \varvec{\beta })= (a^{2^{n-1}})^{{\overline{q}}_n(\overline{\varvec{\alpha }}, \overline{\varvec{\beta }} )}$ with of degree at most n. $\square$

Remark 4.2

It is worth pointing out that the condition $1\ne G_w\subseteq Z$ in Proposition 4.1 can be replaced with $G_w=Z$. Indeed, if $p=2$ then this is clear since $|Z|=2$. If p is odd, the group $G=AB$ is of type (5) and hence here $A\cap B=1$; compare Remark 3.6. Therefore, by the same argument (and using the same notation as) in the proof of Lemma 2.3, it follows that $w({\textbf{b}})=1$ and hence $Z\leqslant A_{v_{w,{\textbf{b}}}}$.

Theorem 4.3

For p a prime, let G be a finite p-group of nilpotency class 2 with a cyclic maximal subgroup. Then G satisfies the Amit–Ashurst conjecture.

Proof

We can assume that G is nonabelian and hence $|G| \geqslant 8$.

By the above classification, we have that G is either the quaternion group $Q_8$ or of type (4) or (5). Since $\langle a^{p^{n-1}}\rangle \leqslant Z(G)$ coincides with the cyclic subgroup Z of A of order p, the quotient is abelian. Now Lemmas 2.2, 2.3 and Proposition 4.1 show that the Amit–Ashurst conjecture holds for G. $\square$

Theorem 4.4

Let G be a maximal class 2-group. Then G satisfies the Amit–Ashurst conjecture.

Proof

A maximal class 2-group is contained in one of the following families: the dihedral groups; the generalised quaternion groups; or the semidihedral groups. Likewise, noting that the quotient G/Z is either of class 2 or isomorphic to a dihedral group, the result follows from Lemmas 2.2, 2.3 and Proposition 4.1. $\square$

Combining Theorems 4.3 and 4.4 proves Theorem A.