AbstractLet w be a word in k variables. For a finite nilpotent group G, a conjecture of Amit states that $$N_w(1)\ge |G|^{k-1}$$
N
w
(
1
)
≥
|
G
|
k
-
1
, where for $$g\in G$$
g
∈
G
, the quantity $$N_w(g)$$
N
w
(
g
)
is the number of k-tuples $$(g_1,\ldots ,g_k)\in G^{(k)}$$
(
g
1
,
…
,
g
k
)
∈
G
(
k
)
such that $$w(g_1,\ldots ,g_k)={g}$$
w
(
g
1
,
…
,
g
k
)
=
g
. Currently, this conjecture is known to be true for groups of nilpotency class 2. Here we consider a generalized version of Amit’s conjecture, which states that $$N_w(g)\ge |G|^{k-1}$$
N
w
(
g
)
≥
|
G
|
k
-
1
for g a w-value in G, and prove that $$N_w(g)\ge |G|^{k-2}$$
N
w
(
g
)
≥
|
G
|
k
-
2
for finite groups G of odd order and nilpotency class 2. If w is a word in two variables, we further show that the generalized Amit conjecture holds for finite groups G of nilpotency class 2. In addition, we use character theory techniques to confirm the generalized Amit conjecture for finite p-groups (p a prime) with two distinct irreducible character degrees and a particular family of words. Finally, we discuss the related group properties of being rational and chiral, and show that every finite group of nilpotency class 2 is rational.
Finite groups with small character degrees and large prime divisors
