International Mathematics Competition
for University Students
2024

IMC2024: Day 1, Problem 4

Problem 4. Let \(\displaystyle g\) and \(\displaystyle h\) be two distinct elements of a group \(\displaystyle G\), and let \(\displaystyle n\) be a positive integer. Consider a sequence \(\displaystyle w=(w_1,w_2,\ldots)\) which is not eventually periodic and where each \(\displaystyle w_i\) is either \(\displaystyle g\) or \(\displaystyle h\). Denote by \(\displaystyle H\) the subgroup of \(\displaystyle G\) generated by all elements of the form \(\displaystyle w_{k}w_{k+1}\ldots w_{k+n-1}\) with \(\displaystyle k\geq 1\). Prove that \(\displaystyle H\) does not depend on the choice of the sequence \(\displaystyle w\) (but may depend on \(\displaystyle n\)).

Ivan Mitrofanov, Saarland University

  Solution