Derived subgroup is characteristic

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This article gives the statement, and possibly proof, of the fact that for any group, the subgroup obtained by applying a given subgroup-defining function (i.e., derived subgroup) always satisfies a particular subgroup property (i.e., characteristic subgroup)}
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Statement

Suppose G is a group. Denote by G' = [G,G] the derived subgroup of G, i.e., the subgroup generated by the commutators of pairs of elements of G. G' is a characteristic subgroup of G.

Related facts

Stronger facts

Proof

Proof idea

The proof idea is that for x,y \in G and \sigma \in \operatorname{Aut}(G):

\sigma([x,y]) = [\sigma(x),\sigma(y)]

Thus, the set of elements expressible as commutators is invariant under any automorphism. Hence, the subgroup generated by this set is also invariant under any automorphism.

Proof details

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