Difference between revisions of "Odd-order cyclic group equals derived subgroup of holomorph"
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==Statement== | ==Statement== | ||
− | Suppose <math>G</math> is 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>. | + | 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== | ==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. | ||
===Corollaries=== | ===Corollaries=== |
Revision as of 23:10, 1 December 2008
Contents
Statement
Suppose is an Odd-order cyclic group (?): a Cyclic group (?) of odd order. Then,
equals the Commutator subgroup (?) of the holomorph
.
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.
Corollaries
- Odd-order cyclic group is fully characteristic in holomorph
- Odd-order cyclic group is characteristic in holomorph
Facts used
- 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
Proof
Given: A cyclic group of odd order.
To prove: equals the commutator subgroup of the holomorph of
: the semidirect product
.
Proof:
-
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
.