D8 in D16
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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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 .
The cosets of are:
The cosets of are:
|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|
Invariance under automorphisms and endomorphisms
|normal subgroup||invariant under inner automorphisms||Yes||index two implies normal|
|characteristic subgroup||invariant under all automorphisms||No||The automorphism interchanges and .|
|fully invariant subgroup||invariant under all endomorphisms||No||(follows from not being characteristic)|
|isomorph-free subgroup||no other isomorphic subgroup||No||The two subgroups are isomorphic.|
|isomorph-automorphic subgroup||all isomorphic subgroups are automorphic subgroups||Yes||are the only subgroups isomorphic to dihedral group:D8.|
|potentially fully invariant subgroup||fully invariant subgroup inside a possibly bigger group||No||See D8 is not potentially fully invariant in D16.|