Group equals Center iff Abelian
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Theorem
Let $G$ be a group.
Then $G$ is abelian if and only if $\map Z G = G$, that is, if and only if $G$ equals its center.
Proof
Necessary Condition
Let $G$ be abelian.
Then:
- $\forall a \in G: \forall x \in G: a x = x a$
Thus:
- $\forall a \in G: a \in \map Z G = G$
$\Box$
Sufficient Condition
Let $\map Z G = G$.
Then by the definition of center:
- $\forall a \in G: \forall x \in G: a x = x a$
and thus $G$ is abelian by definition.
$\blacksquare$
Also see
Sources
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.2$: Groups; the axioms: Examples of groups $\text{(vii)}$
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Entry: conjugacy class