What is a simple example of a finite group-action, preferably on a set, that is free (semi-regular) but not regular/transitive?
Example of free (finite) group action that is not transitive
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abstract-algebra
group-theory
finite-groups
group-actions
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1For a free action which is not transitive : take $X = G' \sqcup G''$ where $G',G''$ are two copies of $G$ and $G$ acts on $X$ with the obvious way. There are two orbits so this is not transitive but this is clear that the action is free. – 2017-02-17
1 Answers
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Let $G$ be any group, and let $H$ be a proper subgroup. Now let $H$ act on $G$ by left translation.