Words and characters in finite p-groups

Abstract

Given a group word w in k variables, a finite group G and g ∈ G, we consider the number N w,G (g) of k-tuples g 1 , . . . , g k of elements of G such that w(g 1 , . . . , g k ) = g. In this work we study the functions N w,G for the class of nilpotent groups of nilpotency class 2. We show that, for the groups in this class, N w,G (1) ≥ |G| k−1 , an inequality that can be improved to N w,G (1) ≥ |G| k /|G w | (G w is the set of values taken by w on G) if G has odd order. This last result is explained by the fact that the functions N w,G are characters of G in this case. For groups of even order, all that can be said is that N w,G is a generalized character, something that is false in general for groups of nilpotency class greater than 2. We characterize group theoretically when N x n ,G is a character if G is a 2-group of nilpotency class 2. Finally we also address the (much harder) problem of studying if N w,G (g) ≥ |G| k−1 for g ∈ G w , proving that this is the case for the free p-groups of nilpotency class 2 and exponent p.

The probability distribution of word maps on finite groups

“…Levy has shown that when G has nilpotent class 2, then for any word w we have µ G,w (1) ≥ 1 |G| [Lev11]; showing that Amit's Conjecture holds for class 2 groups. Iñiguez and Sangroniz have shown the stronger condition that for free p-groups of nilpotency class 2 and exponent 2, it is true that µ G,w (g) ≥ 1 |G| [InS17]. Klyachko and Mkrtchyan [KM14] considered first-order formula in any finite group, which implies µ G,w (1) ≥ 1 |G| when w has only 2 variables.…”

Word problems for finite nilpotent groups

“…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 [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.…”

“…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 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 ,Q8 .…”

“…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 ,Q8 . In [13], the authors also characterize when the function N x n is a character for 2-groups of nilpotency class 2.…”

“…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].…”