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I have not been able to find a proper definition of what an isomorphism class is (in the context of group theory). If one could define it properly for me and give me some help with the following two questions, it will be deeply apprecited:

1) Let $p$ a prime and $X$ a set of cardinality $p+1$. How many isomorphism classes are there of $G-sets$ where $G =\mathbb{Z}/p^2\mathbb{Z}$. What if $G =\mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z}$?

2) List (without repetitions) the isomorphism classes of all transitive actions of $S_4$ on a set of cardinality 3 or 4.

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up vote 2 down vote accepted

Roughly speaking, two actions of a group $G$ on set $X$ are isomorphic if you can relabel elements of $X$ such that one action will turn into the other. You can find a rigorous definition here:

As for isomorphism classes, in your case an isomorphism class of actions of $G$ on $X$ is simply an equivalence class with respect to action isomorphism, i.e. a set of all actions that are isomorphic to a given one.

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