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We say that a group: $(A,*)$ acts on an arbitrary set: $B$ if for all $a \in A$ we can assign a bijection: $a: B \to B$ such that:

1) For all $b \in B$ and $h \in A$ we have that: $g(h(b)) = (gh)(b)$

2) Let $i \in A$ be the identity of $A$, then we have for all $b \in B$: $i(b) = b$.

Is this the same as defining a group action? Or is that something different entirely?

user501050
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    Usually, these two expressions are used interchangeably in the literature. – Levent Mar 15 '18 at 15:14
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    Of course, a group action does not have to be a group acting on a set - often it is a group acting on a set with some additional structure (and the group actions preserve this structure). For example, a group can on itself by right multiplication, left multiplication or by conjugation. Or a group can act on a vector space creating a linear group representation. – gandalf61 Mar 15 '18 at 16:00

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