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18.94 (2014)

Solved

Let $G$ be a group without involutions, $a$ an element of it that is not a square of any element of $G$, and $n$ an odd positive integer. Is it true that the quotient $G/\langle (a^n)^G \rangle$ does not contain involutions?

Contributor: A. I. Sozutov

Progress

No, not always, as shown by an example of V. I. Trofimov; another example: the quotient of $G = \langle a, b \mid a^2 = b^2 \rangle$ by $\langle a^G \rangle$ has order 2.

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