Derived subgroup of dihedral group:D16
From Groupprops
This article is about a particular subgroup in a group, up to equivalence of subgroups (i.e., an isomorphism of groups that induces the corresponding isomorphism of subgroups). The subgroup is (up to isomorphism) cyclic group:Z4 and the group is (up to isomorphism) dihedral group:D16 (see subgroup structure of dihedral group:D16).
The subgroup is a normal subgroup and the quotient group is isomorphic to Klein four-group.
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Contents
Definition
Here, is the dihedral group:D16, the dihedral group of order sixteen (and hence, degree eight). We use here the presentation:
has 16 elements:
The subgroup of interest is the subgroup
. It is cyclic of order 4 and is given by:
The quotient group is a Klein four-group.
Cosets
The subgroup is a normal subgroup, so its left cosets coincide with its right cosets. There are four cosets:
Arithmetic functions
Function | Value | Explanation |
---|---|---|
order of whole group | 16 | |
order of subgroup | 4 | |
index of subgroup | 4 | |
size of conjugacy class (=index of normalizer) | 1 | |
number of conjugacy classes in automorphism class | 1 |
Subgroup-defining functions
Subgroup-defining function | Meaning in general | Why it takes this value |
---|---|---|
derived subgroup | subgroup generated by all commutators | The commutators are precisely the elements of this subgroup. For instance, ![]() |
first agemo subgroup | subgroup generated by all ![]() ![]() |
|
Frattini subgroup | intersection of all maximal subgroups | |
Jacobson radical | intersection of all maximal normal subgroups |
Subgroup properties
Invariance under automorphisms and endomorphisms
Property | Meaning | Satisfied? | Explanation |
---|---|---|---|
normal subgroup | invariant under inner automorphisms | Yes | derived subgroup is normal |
characteristic subgroup | invariant under all automorphisms | Yes | derived subgroup is characteristic |
fully invariant subgroup | invariant under all endomorphisms | Yes | derived subgroup is fully invariant, agemo subgroups are fully invariant |
isomorph-free subgroup | no other isomorphic subgroup | Yes | |
verbal subgroup | generated by set of words | Yes | derived subgroup is verbal, agemo subgroups are verbal |
homomorph-containing subgroup | contains every homomorphic image | No | The subgroup ![]() |
GAP implementation
The group and subgroup can be constructed using GAP's SmallGroup and DerivedSubgroup functions:
G := DihedralGroup(16); H := DerivedSubgroup(G);
The GAP display looks as follows:
gap> G := DihedralGroup(16); H := DerivedSubgroup(G); <pc group of size 16 with 4 generators> Group([ f3, f4 ])Here is a GAP implementation to verify some of the assertions made in this page:[SHOW MORE]