Word problems for finite nilpotent groups

Let $w$ be a word in $k$ variables. For a finite nilpotent group $G$, a conjecture of Amit states that $N_w(1) \ge |G|^{k-1}$, where $N_w(1)$ is the number of $k$-tuples $(g_1,...,g_k)\in G^{(k)}$ such that $w(g_1,...,g_k)=1$. Currently, this conjecture is known to be true for groups of nilpotency class 2. Here we consider a generalized version of Amit's conjecture, and prove that $N_w(g) \ge |G|^{k-2}$, where $g$ is a $w$-value in $G$, for finite groups $G$ of odd order and nilpotency class 2. If $w$ is a word in two variables, we further show that $N_w(g) \ge |G|$, where $g$ is a $w$-value in $G$ for finite groups $G$ of nilpotency class 2. In addition, for $p$ a prime, we show that finite $p$-groups $G$, with two distinct irreducible complex character degrees, satisfy the generalized Amit conjecture for words $w_k =[x_1,y_1]...[x_k,y_k]$ with $k$ a natural number; that is, for $g$ a $w_k$-value in $G$ we have $N_{w_k}(g) \ge |G|^{2k-1}$. Finally, we discuss the related group properties of being rational and chiral, and show that every finite group of nilpotency class 2 is rational.


Introduction
A word w in k variables x 1 , . . . , x k is an element in the free group F k on x 1 , . . . , x k . For any k elements g 1 , . . . , g k in a group G, we can define the element w(g 1 , . . . , g k ) ∈ G by applying to w the group homomorphism from F k to G sending x i to g i , for 1 ≤ i ≤ k.
We denote by G w the set of word values of w in G, i.e. the set of elements g ∈ 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 ∈ G the fibre of g in G (k) is {(g 1 , . . . , g k ) ∈ G (k) | w(g 1 , . . . , g k ) = g}.
If G is a finite group and g ∈ G, we define N w,G (g) to be N w,G (g) = |{(g 1 , . . . , g k ) ∈ G (k) | w(g 1 , . . . , g k ) = g}|; 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 Irr(G), of complex irreducible 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: where N χ w = (N w , χ) = 1 |G| g∈G N w (g)χ(g) = 1 |G| (g 1 ,...,g k )∈G (k) χ w(g 1 , . . . , g k ) is unique for any χ ∈ 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 [9,12,21,1,4,13] and also [10,2,7,17,3,8]. In addition, Nikolov and Segal [18] 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 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, Up till now, Amit's conjecture has only been proved for groups of nilpotency class 2. This was done by Levy [14] 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) ≥ |G| for all finite nilpotent groups G, by Solomon [22]. 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 , . . . , x n ] and for all generalized commutators v n = x 1 x 2 · · · x n x −1 1 x −1 2 · · · x −1 n ; see [20] and [23] 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 ∈ G w , N w (g) ≥ |G| k−1 .
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 [22] we also know that if w is a two variable word, then N w (g) ≥ |G| for all g ∈ Z(G) and all finite nilpotent groups G. Here we improve this result to all groups G of nilpotency class 2 and all g ∈ 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 × K, and g = hk ∈ G w for an n-variable word w with h ∈ H and k ∈ 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 ≤ i ≤ n with h i ∈ H and k i ∈ K, then w(g 1 , . . . , g k ) = w(h 1 k 1 , . . . , h n k n ) = w(h 1 , . . . , h n )w(k 1 , . . . , 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) ≥ |G| for all g ∈ G w .
Iñiguez and Sangroniz [13] proved that the generalized Amit conjecture holds for free p-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) ≥ |G| k−2 for all g ∈ G w .
Next, we extend a result of Pournaki and Sobhani to words w k of the form Pournaki and Sobhani originally considered the single commutator w 1 [19, Theorem 2.2]. Before stating our result, we recall that cd(G) denotes the set of degrees of irreducible complex characters of G, and cs(G) denotes the set of conjugacy class sizes in G.
It is interesting to note, that if instead of requiring |cd(G)| = 2 we require |cs(G)| = 2 then there exist groups with |{N w k (g) : g ∈ G w k }| = n for all positive integers n. This is a recent result due to Naik [16].
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.
We remark that the same result for N w k 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.
Acknowledgements. We thank Dan Segal for many useful conversations. The first author would like to thank the Isaac Newton Trust for supporting her sabbatical during which some of this research took place. The second and third authors thank the Department of Pure Mathematics and Mathematical Statistics of the University of Cambridge, and the third author also thanks Fitzwilliam and Homerton Colleges, for supporting several research visits to Cambridge.

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) → G which sends (x 1 , . . . , x k ) to w(x 1 , . . . , 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 in a group of nilpotency class 2.
Two words w, w ′ ∈ 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 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 Aut(F k ) on F k [13, Proposition 2.3]: 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 ⊆ Z and the result follows from Solomon's result [22]. Now consider words of type (2), so w = x p s 1 1 [x 1 , x 2 ] p s 2 . If G p s 1 ≤ Z then again the result follows from [22]. So, suppose G p s 1 is not central, then Z p s 1 = 1. We now proceed by induction on the order of G, noting that the result holds for abelian groups.
Suppose g ∈ G w and Ω = w −1 (g), the preimage of g in G (2) . Let N = Z p s 1 and consider G = G/N. SetΩ = w −1 (ḡ) ⊆Ḡ (2) . Inductively |Ω| ≥ |Ḡ|. For each v ∈Ω choose a representative (a 1 , a 2 ) ∈ G (2) with (ā 1 ,ā 2 ) = v. Then w(a 1 , a 2 ) = gu p s 1 for some u ∈ Z and then w(a 1 u −1 , a 2 s 2 ) = g for all s 2 ∈ N. So Before proving Theorem B we introduce one more concept, that of the 'defined word map'. We are used to a word w ∈ 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 , ..., a k ) ∈ G (k) , we will write w (i 1 ,...,i k ) (a 1 ,...,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.
We now consider words of type (2). We consider different cases depending on the triviality, or otherwise, of Z p s 1 . When Z p s 1 = 1 we use induction on the order of the group: we assume 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 ∈ G w and in particular g = w(a 1 , . . . , a r ) for some a i ∈ G. Definē Then a −p s 1 1 g =w(a 1 , . . . , a r ). Fixing the odd elements and constructing the corresponding defined word mapw 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 p s 1 1 y p s 1 2 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| ⌈r/2⌉ /|Z|. Thus, combining as before, gives that g has a fibre of size at least (|G| ⌊r/2⌋ /|Z|)(|G| ⌈r/2⌉ /|Z|) = |G| r /|Z| 2 ≥ |G| r−2 .
When Z p s 1 = 1 we proceed by induction and use the usual word map, as seen below.
Here we proceed analogously to the last paragraph in the proof of Theorem A. In the notation of that proof, we obtain  (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.

Characters
Here we show, using character theory techniques, that the generalized Amit conjecture holds for certain words and certain groups .
Recall w k (x 1 , y 1 , . . . , x k , y k ) = [x 1 , y 1 ] · · · [x k , y k ] is the product of k disjoint commutators for k ∈ N, and 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 cd(G) = {1, m} for m > 1.
Proof. For 1 = g ∈ G ′ , using the second orthogonality relation [

Now, noting from [24, Corollary 2.23] that the number of irreducible linear characters is
|G : G ′ |, and from [24,Lemma 2.19] that if χ is linear, then χ(g) = χ(1) since G ′ ≤ ker χ, we obtain 0 = |G : Therefore, Next, as N χ w k = (N w k , χ) = |G| 2k−1 for any χ ∈ Irr(G) by [23, Theorem 1], we have hence the first result. For the final statement, we note that since m ≥ 2, we have Consequently, all elements in G ′ appear as images of w k ; so G ′ = G w k . What is more, since G is non-abelian, we have |G : G ′ | ≥ 2 and hence the lower bound is proved for the fibres.
Proof of Theorem C. By Theorem 3.1, it remains to prove that N w k (1) ≥ |G| 2k−1 and that N w k (1) = N w k (g) for 1 = g ∈ G ′ . For the first part, note that Since we showed in Theorem 3.1 that G ′ = G w k , it follows from the previous result that for . In particular, we note that N w k (1) > |G| 2k |G ′ | > N w k (g) for 1 = g ∈ G ′ , proving there exist exactly two fibre sizes.

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 = G −1 w . 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 [15]. 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 ∈ G and w ∈ F ∞ 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 ∈ 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 ∈ 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 [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. [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 ∈ G and e relatively prime with the order of G.
In particular 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. Proof. First note 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.