D8 in 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) dihedral group:D8 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 cyclic group:Z2.
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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 subgroups and
of interest are:
Both subgroups are normal but are related by the outer automorphism class of .
Cosets
Both subgroups has index two and are normal subgroup (See index two implies normal), so left cosets coincide with right cosets.
The cosets of are:
The cosets of are:
Arithmetic functions
Function | Value | Explanation |
---|---|---|
order of whole group | 16 | |
order of subgroup | 8 | |
index of subgroup | 2 | |
size of conjugacy class (=index of normalizer) | 1 | |
number of conjugacy classes in automorphism class | 2 |
Subgroup properties
Invariance under automorphisms and endomorphisms
Property | Meaning | Satisfied? | Explanation |
---|---|---|---|
normal subgroup | invariant under inner automorphisms | Yes | index two implies normal |
characteristic subgroup | invariant under all automorphisms | No | The automorphism ![]() ![]() ![]() |
fully invariant subgroup | invariant under all endomorphisms | No | (follows from not being characteristic) |
isomorph-free subgroup | no other isomorphic subgroup | No | The two subgroups ![]() |
isomorph-automorphic subgroup | all isomorphic subgroups are automorphic subgroups | Yes | ![]() |
potentially fully invariant subgroup | fully invariant subgroup inside a possibly bigger group | No | See D8 is not potentially fully invariant in D16. |