International Mathematics Competition for University Students 2024

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IMC 2024
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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

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