I'm revising questions on groups for exams, and I still can't quite understand what an Abelian group is. Please help me understand, if anyone could give me a more simple explanation.
An abelian group $G$ is a group $G$ such that the order of multiplication doesn't matter. Precisely: an abelian group is such that $ab = ba$ for all $a,b \in G$.
An example of an abelian group: the integers.
A non-example: the group $S_3$ of permutations on 3 letters.
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A group is abelian iff its irreducible representation $\rho$ has dimension 1.
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