Odd-order cyclic group equals derived subgroup of holomorph
- Odd-order cyclic group is fully characteristic in holomorph
- Odd-order cyclic group is characteristic in holomorph
- Cyclic implies Aut-Abelian: The automorphism group of a cyclic group is Abelian.
- Inverse map is automorphism iff Abelian: For an Abelian group, the map sending every element to its inverse is an automorphism.
- kth power map is bijective iff k is relatively prime to the order
Given: A cyclic group of odd order.
To prove: equals the commutator subgroup of the holomorph of : the semidirect product .
- contains the commutator subgroup of : By fact (1), is Abelian, so is Abelian. Thus, contains the commutator subgroup .
- The commutator subgroup of contains : For this, note (fact (2)) that the inverse map is an automorphism of , say, denoted by an element . The commutator between any and is , so the set of squares of elements of is in . By fact (3), every element of is the square of an element of , so is contained in .