In this, involutions in a finite group are either conjugate or have an involution centralizing both of them. I wonder if there are similar results for an infinite group. I think and look for it but I can't finish it. When I read this proof, I wonder if a product of two involutions can be written as a commutator. Thanks for all your support.
2026-03-26 01:26:38.1774488398
Involution elements in infinite groups
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Let $x,y$ in a group satisfy $x^2=y^2=1$. Then (as David Craven wrote in an erased comment) one has one of the following:
Indeed, if $xy$ has finite odd order, then (1) holds. If $xy$ has finite even order, then (2) holds. And otherwise (3) holds.
In addition, no condition can be removed, as generators in various dihedral groups show. Also, in suitable groups and pairs, any two of these conditions can be satisfied without the third, or all three simultaneously.