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Suppose <math>G</math> is an [[fact about::odd-order cyclic group]]: a [[fact about::cyclic group]] of odd order. Then, <math>G</math> equals the [[fact about::commutator subgroup]] of the [[fact about::holomorph of a group|holomorph]] <math>G \rtimes \operatorname{Aut}(G)</math>.
==Related facts==
===Breakdown at the prime two===
The analogous statement is not true for all groups of even order. In fact, the commutator subgroup of a cyclic group of even order is the subgroup comprising the squares in that group, which has index two in the group.
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