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Definition
Let \(G\) be a group and let \(S\) be a set. A multiplication of elements of \(S\) by elements of \(G\) (defined by a function from $G × S → $) is called a group action of \(G\) on \(S\) provided for each \(x \in S\):
- \(a(bx) = (ab)x ~\forall a,b \in G\), and
- \(ex = x\) for the identity element \(e\) of \(G\).