Let $X$ $=$ $\{$$1$, $2$, $3$$\}$ and $G$ $=$ $\mathbb Z_2$. How many different G-actions are there on $X$?
Just learned group action. Need some hint on this one.
Thanks.
2026-04-03 17:38:47.1775237927
The number of different G-actions on X
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Hint: prove any such group action either does nothing or has the nontrivial group element switch two things (and leave the third fixed).
With more algebra: a group action will be a homomorphism $\mathbb{Z}_2\to S_3$, which will be determined by where it sends the nontrivial element, which must be taken to either the identity or an element of order two. You can list out all the elements of $S_3$ and identify the elements of order two.