MathDB

Subgroup generated by length-n words of an infinite g-h-sequence.

Source: IMC 2024, Problem 4

August 7, 2024

group theoryabstract algebraCombinatorics of wordssubgroupSequences

Problem Statement

Let

g

and

h

be two distinct elements of a group

G

, and let

n

be a positive integer. Consider a sequence

g

which is not eventually periodic and where each

w_i

is either

g

or

h

. Denote by

H

the subgroup of

G

generated by all elements of the form

w_kw_{k+1}\dotsc w_{k+n-1}

with

k \ge 1

. Prove that

H

does not depend on the choice of the sequence

w

(but may depend on

n

).

Back to Problems View on AoPS