1 Introduction

A word w in k variables \(x_1,\ldots ,x_k\) is an element in the free group \(F_k\) on \(x_1,\ldots ,x_k\). For any k elements \(g_1,\ldots ,g_k\) in a group G, we can define the element \(w(g_1,\ldots ,g_k)\in G\) by applying to w the group homomorphism from \(F_k\) to G sending \(x_i\) to \(g_i\) for \(1\le i\le k\).

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 \(G^{(k)}\), the direct product of k copies of G.

For a word w in k variables and a group G, for any \(g\in G\), the fibre of g in \(G^{(k)}\) is

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

If G is a finite group and \(g\in G\), we define \(N_{w,G}(g)\) to be

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

i.e., the size of the fibre of g in \(G^{(k)}\). When the group G is clear, we will simply write \(N_w(g)\). The function \(N_{w,G}\) is a (non-negative) integer-valued class function since it is constant on the conjugacy classes. The set \({\text {Irr}}(G)\) of irreducible complex characters of G is an orthonormal basis for the vector space of the complex class functions and \(N_{w,G}\) can be written as a linear combination of the irreducible characters of G:

$$\begin{aligned} N_{w,G}=N_w=\sum _{\chi \in {\text {Irr}}(G)}N_w^{\chi }\,\chi \end{aligned}$$

where

$$\begin{aligned} N_{w}^{\chi }=(N_w,\chi )=\frac{1}{|G|}\sum _{g\in G}N_w(g)\overline{\chi (g)}=\frac{1}{|G|}\sum _{(g_1,\ldots ,g_k)\in G^{(k)}}\overline{ \chi \big (w(g_1,\ldots ,g_k)\big )} \end{aligned}$$

is unique for any \(\chi \in {\text {Irr}}(G)\).

Much about the functions \(N_{w,G}\), or rather \(P_{w,G}=N_{w,G}/|G|^k\), has been done, particularly for the commutator word \(w=[x,y]\) and for the case G is a p-group for some prime p; see [1, 4, 9, 12, 13, 22] and also [2, 3, 7, 8, 10, 18]. In addition, Nikolov and Segal [20] gave a characterization of finite nilpotent groups and of finite solvable groups based on the function \(P_{w,G}\): a finite group is nilpotent if and only if the values of \(P_{w,G}(g)\) are bounded away from zero as g ranges over \(G_w\) and w ranges over all group words; and a finite group is solvable if and only if the probabilities \(P_{w,G}(1)\) are bounded away from zero as w ranges over all group words. Iñiguez and Sangroniz [13] proved that for any finite group G of nilpotency class 2 and any word w, the function \(N_w\) is a generalized character of G, that is, a \({\mathbb {Z}}\)-linear combination of irreducible characters. What is more, if G is a finite p-group of nilpotency class 2 with p odd and w any word, then \(N_w\) is a character of G. In general, for \(p=2\), the function \(N_w\) is not a character; one can easily check for \(N_{x^2,Q_8}\). In [13], the authors also characterize when the function \(N_{x^n}\) is a character for 2-groups of nilpotency class 2.

The following is well known; see [1]:

Conjecture

(Amit). For any finite nilpotent group G and any word w in k variables,

$$\begin{aligned} N_w(1)\ge |G|^{k-1}. \end{aligned}$$

Up till now, Amit’s conjecture has only been proved for groups of nilpotency class 2. This was done by Levy [15] and independently by Iñiguez and Sangroniz [13].

Amit’s conjecture is seen to hold for certain words w. If w is a two-variable word, then \(N_w(1)\ge |G|\) for all finite nilpotent groups G by Solomon [23]. Whenever \(N_{w,G}\) is a character, Amit’s conjecture also holds; see [13]. It then follows that Amit’s conjecture holds for all left-normed commutators \(w_n=[x_1,\ldots , x_n]\) and for all generalized commutators \(v_n=x_1x_2\cdots x_nx_1^{-1}x_2^{-1}\cdots x_n^{-1}\); see [19, 24] respectively.

We consider a version of Amit’s conjecture as applied to general fibres:

Conjecture

(Generalized Amit conjecture). For any finite nilpotent group G, any word w in k variables, and any \(g\in G_w\),

$$\begin{aligned} N_w(g)\ge |G|^{k-1}. \end{aligned}$$

This appears as a conjecture in Ashurst’s thesis [4, Conjecture 6.2.1]. Note that the bound \(N_w(g)=|G|^{k-1}\) is achieved by a surjective word map with uniform distribution for example \(w=x[y,z]\). Moreover, Cocke and Ho have shown that a finite group is nilpotent if and only if every surjective word map has uniform distribution [6, Theorem B], so the Amit bound is met by all surjective word maps. By Solomon’s result in [23], we also know that if w is a two variable word, then \(N_w(g)\ge |G|\) for all \(g \in Z(G)\) and all finite nilpotent groups G. Here we improve this result to all groups G of nilpotency class 2 and all \(g \in G_w\).

However first note that since a finite nilpotent group is a direct product of its Sylow subgroups, it suffices to consider finite p-groups. This is because if \(G=H\times K\), and \(g=hk\in G_w\) for an n-variable word w with \(h\in H\) and \(k\in K\), then \(N_{w,G}(g) = N_{w,H}(h) N_{w,K}(k)\). This relies on the fact that if \(g_i=h_i k_i\) for \(1\le i\le n\) with \(h_i\in H\) and \(k_i\in K\), then \(w(g_1,\ldots ,g_k)=w(h_1k_1,\ldots ,h_nk_n)=w(h_1,\ldots , h_n)w(k_1,\ldots , k_n)\). Hence if the conjecture holds for each group H and K, it then holds for their direct product G.

Theorem A

Suppose G is a finite p-group of nilpotency class 2 and w is a word in two variables. Then \(N_w(g) \ge |G|\) for all \(g \in G_{w}\).

Iñiguez and Sangroniz [13] proved that the generalized Amit conjecture holds for free \(p\text{-groups }\) of nilpotency class 2 and exponent p. Our next result does not meet Amit’s bound, but can be proved for all words w and for all groups G of odd order and nilpotency class 2.

Theorem B

Suppose G is a finite p-group of nilpotency class 2 for p an odd prime, and w is a word in k variables. Then \(N_w(g) \ge |G|^{k-2}\) for all \(g \in G_w\).

Next, we extend a result of Pournaki and Sobhani to words \(w_{\ell }\) of the form\(w_{\ell }=[x_1, y_1] \cdots [x_{\ell },y_{\ell }]\). Pournaki and Sobhani originally considered the single commutator \(w_1\) [21, Theorem 2.2]. Before stating our result, we recall that \(\mathrm{cd}(G)\) denotes the set of degrees of irreducible complex characters of G, and \(\mathrm{cs}(G)\) denotes the set of conjugacy class sizes in G.

Theorem C

Let G be a finite p-group such that \(\mathrm{cd}(G) = \{1,m\}\) for \(m >1\) and \(w_{\ell }=[x_1, y_1] \cdots [x_{\ell },y_{\ell }]\) a product of \({\ell }\) disjoint commutators for \({\ell }\in {\mathbb {N}}\). Then \(G'=G_{w_{\ell }}\) and \(|\{N_{w_{\ell }}(g): g \in G'\}|=2\). Furthermore \(N_{w_{\ell }}(g) \ge |G|^{2{\ell }-1}\) for all \(g\in G'\).

It is interesting to note that if instead of requiring \(|\mathrm{cd}(G)| =2\) we require \(|\mathrm{cs}(G)| =2\), then there exist groups with \(|\{N_{w_{\ell }}(g): g \in G_{w_{\ell }}\}|=n\) for all positive integers n. This is a recent result due to Naik [17].

Theorem C yields the following corollary.

Corollary D

Let G be a finite group of nilpotency class 2 and \(|G'|=p\) with p a prime. Then \(N_{w_{\ell }}(g) \ge |G|^{2{\ell }-1}\) whenever \(g \in G'\).

We remark that the same result for \(N_{w_{\ell }}\) was obtained in [13, Propositions 6.1 and 6.2] for finite p-groups with different restrictions.

Finally we consider the different notions of rationality and chirality; see Section 4 for definitions. In particular, we point out that if G is a finite group of nilpotency class 2 and w is a word, then \(N_w(g) = N_w(g^e)\) for all e coprime to the order of G. This is an improvement on [5, Theorem 5.2] for the case of finite groups.

All groups in this paper are finite.

2 Fibres of non-identity elements

Given a group G of nilpotency class 2 and a word w, we consider the sizes of fibres of non-identity elements under the word map.

First we observe that if the word map \(w:G^{(k)} \rightarrow G\) which sends \((x_1, \ldots , x_k)\) to \(w(x_1, \ldots , x_k)\) is a homomorphism, then the fibre of any element in \(G_w\) is a coset of the kernel of the map. Hence the fibre of each element in \(G_w\) is of the same size, namely \(|G|^k/|G_w|\) (which is at least \(|G|^{k-1}\)). When G is abelian, all word maps are homomorphisms. We use this idea to analyse the nilpotency class 2 case.

Another key observation, that will be used throughout, is that \([xy,z] = [x,z][y,z]\) and \([x,yz]=[x,y][x,z]\) in a group of nilpotency class 2.

Two words \(w, w' \in F_k\) are said to be equivalent if they belong to the same orbit under the action of the automorphism group of \(F_k\). In [13, Proposition 2.1], the authors prove that if w and \(w'\) are equivalent, then \(N_{w,G} = N_{w',G}\) for any finite p-group G of nilpotency class 2. They then go on to prove that the following words are a system of representatives of the action of \(\mathrm{Aut}(F_k)\) on \(F_k\) [13, Proposition 2.3]:

$$\begin{aligned}&{[}x_1, x_2]^{p^{s_1}} \cdots [x_{2r-1}, x_{2r}]^{p^{s_r}}, \quad \text {for }0 \le s_1 \le \cdots \le s_r, \end{aligned}$$
(1)
$$\begin{aligned}&x_1^{p^{s_1}} [x_1, x_2]^{p^{s_2}} [x_2, x_3]^{p^{s_3}} \cdots [x_{r-1}, x_r]^{p^{s_r}}, \quad \text {for } s_1 \ge 0,\; 0 \le s_2 \le \cdots \le s_r. \end{aligned}$$
(2)

Thus, it is enough for us to consider words of these types. We can now prove Theorem A.

Proof of Theorem A

Let Z denote the centre of G. We first consider words of type (1), so \(w = [x_1, x_2]^{p^{s_1}}\). In this case, \(G_w \subseteq Z\) and the result follows from Solomon’s result [23].

Now consider words of type (2), so \(w = x_1^{p^{s_1}}[x_1, x_2]^{p^{s_2}}\). If \(G^{p^{s_1}} \le Z\), then again the result follows from [23]. So, suppose \(G^{p^{s_1}}\) is not central, then \(Z^{p^{s_1}} \ne 1\). We now proceed by induction on the order of G, noting that the result holds for abelian groups.

Suppose \(g \in G_w\) and \(\varvec{\Omega } = w^{-1}(g)\), the preimage of g in \(G^{(2)}\). Let \(N = Z^{p^{s_1}}\) and consider \({\bar{G}} = G/N\). Set \(\bar{\varvec{\Omega }} = w^{-1}({\bar{g}}) \subseteq {\bar{G}}^{(2)}\). Inductively \(|\bar{\varvec{\Omega }}| \ge |{\bar{G}}|\). For each \(\mathbf{v} \in \bar{\varvec{\Omega }}\), choose a representative \((a_1, a_2) \in G^{(2)}\) with \(({\bar{a}}_1, {\bar{a}}_2) = \mathbf{v}\). Then \(w(a_1, a_2) = gu^{p^{s_1}}\) for some \(u \in Z\) and then \(w(a_1u^{-1}, a_2 s_2) = g\) for all \(s_2 \in N\). So

$$\begin{aligned} \varvec{\Omega } \supseteq \bigcup _{\mathbf{v} \in \bar{\varvec{\Omega }}} \{a_1 u^{-1}\} \times a_2 N, \end{aligned}$$

a disjoint union. Hence \(|\varvec{\Omega }| \ge |N| |\bar{\varvec{\Omega }} |\ge |G|\). \(\square \)

Before proving Theorem B, we introduce one more concept, that of the ‘defined word map’. We are used to a word \(w \in F_k\) defining a word map from \(G^{(k)}\) to G. In a defined word map, some of the entries are fixed elements of G and are not allowed to vary. For a fixed tuple \((a_1,\ldots , a_k)\in G^{(k)}\), we will write \(w^{(i_1,\ldots , i_k)}_{(a_1,\ldots , a_k)}\) for the defined word map where the \( i_j\)-th term is replaced with \(a_j\). This is particularly useful when G is of nilpotency class 2 as then this defined word map is often a homomorphism.

Proof of Theorem B

We first consider words of type (1), in this case, the argument also works for \(p=2\). Given w of type (1), we consider the corresponding defined word map given by fixing the even entries as \((a_2, a_4, \ldots , a_{2r}) \in G^{(r)}\) say. That is

$$\begin{aligned}&w^{(2,4,\ldots , 2r)}_{(a_2, a_4, \ldots , a_{2r})}: G \times \overset{r}{\cdots }\times G \rightarrow G,\\&(x_1, x_3, \ldots , x_{2r-1}) \mapsto [x_1, a_2]^{p^{s_1}} \cdots [x_{2r-1}, a_{2r}]^{p^{s_r}}. \end{aligned}$$

As G is of nilpotency class 2, this map is a homomorphism. Furthermore, the image is a subgroup of the centre Z of G and thus the kernel has size at least \(|G|^{r}/|Z|\).

Suppose \(g \in G_w\) and in particular \(g = w(a_1, a_2, a_3 \ldots , a_{2r})\) for some \(a_i \in G\). Now, by the previous paragraph, if we fix the \(a_i\) for i even, we see there are at least \(|G|^{r}/|Z|\) tuples \((b_1, b_3, \ldots , b_{2r-1})\) satisfying \(g = w(b_1, a_2, b_3, \ldots , b_{2r-1}, a_{2r})\). Fix such a \((b_1, b_3, \ldots , b_{2r-1})\), and construct the new defined word map \(w^{(1,3,\ldots , 2r-1)}_{(b_1, b_3, \ldots , b_{2r-1})}\) which sends \(G^{(r)}\) to G by mapping \((x_2, x_4, \ldots , x_{2r})\) to \(w(b_1, x_2, \ldots , b_{2r-1}, x_{2r})\). Again this is a homomorphism to Z. Note that g lies in the image of each of these maps, and the preimage of g for each of these homomorphisms has size at least \(|G|^r/|Z|\). Thus summing over the r-tuples \((b_1, b_3, \ldots , b_{2r-1})\) yields that \(N_w(g) \ge (|G|^r/|Z|)(|G|^r/|Z|) = |G|^{2r}/|Z|^2 \ge |G|^{2r-2}\).

We now consider words of type (2). We consider different cases depending on whether \(Z^{p^{s_1}}\) is trivial or not. When \(Z^{p^{s_1}} \ne 1\), we use induction on the order of the group: we assume that for all groups of smaller order of nilpotency class 2 or less, our result holds. Note the result holds for abelian groups so we have the base step.

Case (i): Suppose \(Z^{p^{s_1}}=1\).

Suppose \(g \in G_w\) and in particular \(g = w(a_1, \ldots , a_r)\) for some \(a_i \in G\). Define

$$\begin{aligned} {\bar{w}}=[x_1, x_2]^{p^{s_2}}[x_2, x_3]^{p^{s_3}} \cdots [x_{r-1}, x_r]^{p^{s_r}}. \end{aligned}$$

Then \(a_1^{-p^{s_1}}g = {\bar{w}}(a_1, \ldots , a_r)\). Fixing the odd elements and constructing the corresponding defined word map \({\bar{w}}^{(1,3,\ldots , t)}_{(a_1, a_3, \ldots , a_t)}\), where \(t = 2\lceil r/2 \rceil -1\), gives a homomorphism into Z. Thus the number of tuples \((b_2, \ldots , b_s)\), with \(s=2 \lfloor r/2 \rfloor \), such that \({\bar{w}}(a_1, b_2, \ldots ) = a_1^{-p^{s_1}}g\) and thus \(w(a_1, b_2, \ldots ) =g\) is at least \(|G|^{\lfloor r/2 \rfloor }/|Z|\).

Fixing the even elements of w does define a homomorphism as we have insisted \(Z^{p^{s_1}}=1\) and p odd and thus \((y_1 y_2)^{p^{s_1}} = y_1^{p^{s_1}}y_2^{p^{s_1}}\) as required. Furthermore \(G^{p^{s_1}}\) is central, as \(Z^{p^{s_1}} = 1\), and thus the corresponding defined word map is a homomorphism into Z. So, fixing the even elements yields a homomorphism with kernel of order at least \(|G|^{\lceil r/2 \rceil }/|Z|\). Thus, combining as before gives that g has a fibre of size at least

$$\begin{aligned} (|G|^{\lfloor r/2 \rfloor }/|Z|)(|G|^{\lceil r/2 \rceil }/|Z|)= |G|^r/|Z|^2 \ge |G|^{r-2}. \end{aligned}$$

When \(Z^{p^{s_1}} \ne 1\), we proceed by induction and use the usual word map, as seen below.

Case (ii): Suppose \(Z^{p^{s_1}} \ne 1\).

Here we proceed analogously to the last paragraph in the proof of Theorem A. In the notation of that proof, we obtain

$$\begin{aligned} \varvec{\Omega } \supseteq \bigcup _{\mathbf{v} \in \bar{\varvec{\Omega }}} \{ a_1u^{-1}\} \times a_2N \times \cdots \times a_rN, \end{aligned}$$

a disjoint union. Thus \(|\varvec{\Omega }| \ge |N|^{r-1} |\bar{\varvec{\Omega }}| \ge |G|^{r-2}\). \(\square \)

Remark 2.1

(i) For a word w of type (1) and all primes p, if \(|Z|^2\le |G|\), then the generalized Amit conjecture holds.

(ii) For a word w of type (2) with \(s_1=0\), the word map defined by w is surjective and hence its distribution is uniform; cf. [4, Lemma 3.2.1] or [6, Theorem B]. This implies that the generalized Amit conjecture holds.

3 Characters

Here we show, using character theory techniques, that the generalized Amit conjecture holds for certain words and certain groups.

Recall \(w_{\ell }(x_1, y_1, \ldots , x_{\ell }, y_{\ell }) = [x_1, y_1] \cdots [x_{\ell }, y_{\ell }]\) is the product of \({\ell }\) disjoint commutators for \({\ell }\in {\mathbb {N}}\), and \(\mathrm{cd}(G)\) is the set of degrees of irreducible complex characters of G. The following results first appeared in the second author’s thesis.

Theorem 3.1

Let G be a finite p-group such that \(\mathrm{cd}(G) = \{1,m\}\) for \(m >1\). If \(1 \ne g \in G'\), then

$$\begin{aligned} N_{w_{\ell }}(g) = \frac{|G|^{2{\ell }}}{|G'|} \Big (1 - \frac{1}{m^{2{\ell }}}\Big ). \end{aligned}$$

Furthermore \(G'=G_{w_{\ell }}\) and \(N_{w_{\ell }}(g) \ge |G|^{2{\ell }-1}\) for \(1\ne g\in G'\).

Proof

For \(1\ne g\in G'\), using the second orthogonality relation [14, Theorem 2.18], we have

$$\begin{aligned} 0&= \sum _{\chi \in \text {Irr}(G)}\chi (g)\chi (1)=\sum _{{ \chi \in \text {Irr}(G)}\atop {\chi (1)=1}} \chi (g) +\sum _{{ \chi \in \text {Irr}(G)}\atop {\chi (1)=m}} \chi (g)\chi (1) \\&=\sum _{{ \chi \in \text {Irr}(G)}\atop {\chi (1)=1}} \chi (g) +m\sum _{{ \chi \in \text {Irr}(G)}\atop {\chi (1)=m}} \chi (g). \end{aligned}$$

Now, noting from [14, Corollary 2.23] that the number of irreducible linear characters is \(|G:G'|\), and from [14, Lemma 2.19] that if \(\chi \) is linear, then \(\chi (g)=\chi (1)\) since \(G'\le \ker \chi \), we obtain

$$\begin{aligned} 0=|G:G'| + m\sum _{{ \chi \in \text {Irr}(G)}\atop {\chi (1)=m}} \chi (g). \end{aligned}$$

Therefore,

$$\begin{aligned} \sum _{{ \chi \in \text {Irr}(G)}\atop {\chi (1)=m}} \chi (g)=-\frac{|G:G'|}{m}. \end{aligned}$$

Next, as \(N_{w_{\ell }}^{\chi }=(N_{w_{\ell }},\chi )=\big (\frac{|G|}{\chi (1)}\big )^{2{\ell }-1}\) for any \(\chi \in \text {Irr}(G)\) by [24, Theorem 1], we have

$$\begin{aligned} N_{w_{\ell }}(g)&=\sum _{\chi \in \text {Irr}(G)}\Big (\frac{|G|}{\chi (1)}\Big )^{2{\ell }-1}\cdot \chi (g)\\&=\sum _{{ \chi \in \text {Irr}(G)}\atop {\chi (1)=1}} |G|^{2{\ell }-1} +\sum _{{ \chi \in \text {Irr}(G)}\atop {\chi (1)=m}} \Big (\frac{|G|}{m}\Big )^{2{\ell }-1}\cdot \chi (g)\\&=|G|^{2{\ell }-1}\cdot |G:G'| +\Big (\frac{|G|}{m}\Big )^{2{\ell }-1}\cdot \Big (\frac{-|G:G'|}{m}\Big )\\&=\frac{|G|^{2{\ell }}}{|G'|}\cdot \Big (1-\frac{1}{m^{2{\ell }}}\Big ), \end{aligned}$$

hence the first result. For the final statement, we note that since \(m\ge 2\), we have

$$\begin{aligned} \Big (1-\frac{1}{m^{2{\ell }}}\Big )\ge \frac{3}{4}~. \end{aligned}$$

Consequently, all elements in \(G'\) appear as images of \(w_{\ell }\); so \(G'=G_{w_{\ell }}\). What is more, since G is non-abelian, we have \(|G:G'|\ge 2\) and hence the lower bound is proved for the fibres. \(\square \)

Proof of Theorem C

By Theorem 3.1, it remains to prove that \(N_{w_{\ell }}(1)\ge |G|^{2{\ell }-1}\) and that \(N_{w_{\ell }}(1)\ne N_{w_{\ell }}(g)\) for \(1\ne g\in G'\). For the first part, note that

$$\begin{aligned} N_{w_{\ell }}(1)+\sum _{1\ne g\in G'} N_{w_{\ell }}(g) =\sum _{g\in G'} N_{w_{\ell }}(g) = |G|^{2{\ell }}. \end{aligned}$$
(3)

Since we showed in Theorem 3.1 that \(G'=G_{w_{\ell }}\), it follows from the previous result that for \(1\ne g\in G'\),

$$\begin{aligned} N_{w_{\ell }}(g) = \frac{|G|^{2{\ell }}}{|G'|}\Big (1-\frac{1}{m^{2{\ell }}}\Big ) < \frac{|G|^{2{\ell }}}{|G'|} \end{aligned}$$

and hence \(N_{w_{\ell }}(1)>\frac{|G|^{2{\ell }}}{|G'|}> |G|^{2{\ell }-1}\) using (3).

In particular, we note that \(N_{w_{\ell }}(1)>\frac{|G|^{2{\ell }}}{|G'|}> N_{w_{\ell }}(g)\) for \(1\ne g\in G'\), proving there exist exactly two fibre sizes. \(\square \)

Proof of Corollary D

For \(g=1\), the result is true by [13, 15]. We claim that a non-linear irreducible character \(\chi \) vanishes outside of the centre Z of G. Consider \(g\in G\backslash Z\). So there exists some \(x\in G\) such that \(t=[g,x]\ne 1\). Since \(|G'|=p\), the element t is a generator of \(G'\). If we now consider a complex representation \(\rho \) affording \(\chi \), we have that \(\rho (t)=\epsilon I\) where \(\epsilon \in {\mathbb {C}}\) by [14, Lemma 2.25]. In the case \(\epsilon =1\), we have \(t\in \ker \rho \) and therefore \(G'\le \ker \rho \) which is a contradiction to \(\chi \) being non-linear; compare [14, Lemma 2.22]. Therefore \(\epsilon \ne 1\) and since

$$\begin{aligned} \chi (g)=\chi (g^{x})=\chi (gt)=\text {tr}_{\rho }(gt)=\text {tr}(\rho (g)\rho (t))=\text {tr}(\epsilon \rho (g)I)=\epsilon \,\chi (g), \end{aligned}$$

we conclude that \(\chi (g)=0\), and the claim holds. From [14, Corollary 2.28 and Lemma 2.29], we deduce that \(\chi (1)^2=|G:Z|\). Therefore G is a group of central type with just two irreducible complex character degrees, i.e., \(\mathrm{cd}(G) = \{1,|G : Z|^{1/2}\}\). Now the assertion holds using the previous theorem. \(\square \)

4 Rationality and chirality

In this section, we draw together some definitions and ideas that have appeared in the literature and conclude with a corollary which, although is a direct consequence of the results of [13], has not previously been explicitly stated and we believe is of interest.

According to [5], a pair (G, w), where G is a group and w is a word, is called chiral if \(G_w\ne G_w^{-1}\). The group G is called chiral if (G, w) is chiral for some w. Otherwise G is achiral. In [5], the authors comment that the existence of chiral groups follows from a result of Lubotzky [16]. They then began the process of classifying all finite chiral groups. In particular, they found all chiral groups of order less than 108; there are two of them. These results negatively answer a question posed by Ashurst in her thesis [4, Question 5]: If G is a finite group, \(g \in G\), and \(w \in F_{\infty }\), is it necessarily true that \(P(G, w=g) = P(G, w=g^{-1})\)?

Related to the definition of achiral is the definition of weakly rational. According to [11], a word w is rational if for every finite group G and any \(g\in G\), we have \(N_w(g)=N_w(g^{e})\) for every e relatively prime to |G|. Additionally, a word w is weakly rational if and only if for every finite group G and for every integer e relatively prime to |G|, the set \(G_w\) is closed under e-th powers. Clearly rational implies weakly rational; see [11] for more discussion.

We change the emphasis of the definition and say a pair (G, w) for G a group and w a word is rational if for all \(g \in G\) and for every e relatively prime to |G|, we have \(N_w(g)=N_w(g^e)\). A group G is rational if (G, w) is rational for all words w. Similarly we define a pair (G, w) to be weakly rational if for every e relatively prime to |G|, the set \(G_w\) is closed under e-th powers. A group G is weakly rational if (G, w) is weakly rational for all pairs (G, w) running over all words w. Clearly if G is rational, it is weakly rational and if it is weakly rational, it is achiral.

In [2] and [13], the authors show that the pair (G, w) is rational if and only if \(N_w\) is a generalized character of G.

Lemma 4.1

( [2, Corollary 3.3] and [13, Lemma 3.1]). Let G be a group and w a word. Then \(N_w=N_{w,G}\) is a generalized character of G if and only if \(N_w(g)=N_w(g^{e})\) for any \(g\in G\) and e relatively prime to the order of G.

In particular, in [13], they showed this is exactly what happens for any word w and any finite group of nilpotency class 2.

Theorem 4.2

( [13, Theorem 3.2]). Let G be a p-group of nilpotency class 2 and w a word. Then \(N_w=N_{w,G}\) is a generalized character of G.

We include these results here to highlight the following corollary which is a partial improvement on [5, Theorem 5.2] which says that all class 2, rank 3, nilpotent groups are achiral.

Corollary 4.3

Every finite group G of nilpotency class 2 is rational.

Proof

First note that an abelian group is rational, and by Lemma 4.1 and Theorem 4.2, a p-group of nilpotency class 2 is rational. It follows for finite nilpotent groups of class 2 using the comment before Theorem A. \(\square \)