A generalization of a question asked by B. H. Neumann

Publication date: 28 January 2022

Abstract: Let

w i n F_{2}

be a word and let

m

and

n

be two positive integers. We say that a finite group

G

has the

w_{m, n}

-property if however a set

M

of

m

elements and a set

N

of

n

elements of the group is chosen, there exist at least one element of

x i n M

and at least one element of

y i n M

such that

w (x, y) = 1 .

Assume that there exists a constant

g a m m a < 1

such that whenever

w

is not an identity in a finite group

X

, then the probability that

w (x_{1}, x_{2}) = 1

in

X

is at most

g a m m a .

If

m l e q n

and

G

satisfies the

w_{m, n}

-property, then either

w

is an identity in

G

or

| G |

is bounded in terms of

g a m m a, m

and

n

. We apply this result to the 2-Engel word.

This page was built for publication: A generalization of a question asked by B. H. Neumann