Suppose $G$ is a group and $G = \bigcup_{a \in A} \langle a \rangle B$ for finite sets $A, B \subset G$. Is $G$ virtually cyclic?
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what is "virtually" cyclic? – infinity Mar 13 '20 at 12:00
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@infinity A group is virtually cyclic if it contains a cyclic subgroup of finite index. – Mar 13 '20 at 12:01
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@Gae.S. thanks, didn't know that. – infinity Mar 13 '20 at 12:04
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Yes. See here – Derek Holt Mar 13 '20 at 12:18
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Thanks, I actually just came up with a proof but I found this surprisingly puzzling. – Ville Salo Mar 13 '20 at 12:19
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My mistake was trying to get a slick algebraic proof for this, dynamically it's more or less obvious, color the group according to the cosets and take a minimal subsystem of the orbit closure. – Ville Salo Mar 13 '20 at 12:37