D8 in SD16
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) semidihedral group:SD16 (see subgroup structure of semidihedral group:SD16).
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 semidihedral group:SD16, the semidihedral 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 dihedral of order 8 and is given by:
The multiplication table of is as follows:
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Cosets
The subgroup has index two and is hence a normal subgroup (See index two implies normal). Its left cosets coincide with its right cosets. There are two cosets:
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 | 1 | |
size of automorphism class of subgroup | 1 |
Subgroup-defining functions
Subgroup-defining function | Meaning in general | Why it takes this value |
---|---|---|
first omega subgroup | subgroup generated by elements of order two | The elements of order two are ![]() |
join of abelian subgroups of maximum rank | subgroup generated by the abelian subgroups that have maximum rank | There are no rank 3 abelian subgroups, and the rank 2 abelian subgroups are ![]() ![]() |
join of elementary abelian subgroups of maximum order | subgroup generated by the elementary abelian subgroups of maximum order | There are no elementary abelian subgroups of order 8, and the elementary abelian subgroups of order 4 are ![]() ![]() |
Subgroup properties
Invariance under automorphisms and endomorphisms
Property | Meaning | Satisfied? | Explanation |
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normal subgroup | invariant under inner automorphisms | Yes | index two implies normal |
characteristic subgroup | invariant under all automorphisms | Yes | On account of being the first omega subgroup, also on account of being an isomorph-free subgroup. |
fully invariant subgroup | invariant under all endomorphisms | Yes | On account of being the first agemo subgroup. |
isomorph-free subgroup | no other isomorphic subgroup | Yes | The other two subgroups of order 8 are Z8 in D16 and Q8 in D16, both non-isomorphic to it. Also follows from being the first omega subgroup. |
homomorph-containing subgroup | contains every homomorphic image | Yes | On account of being an omega subgroup. |
image-closed characteristic subgroup | under any surjective homomorphism from ![]() ![]() ![]() ![]() |
No | If we consider ![]() ![]() |
image-closed fully invariant subgroup | under any surjective homomorphism from ![]() ![]() ![]() ![]() |
No | Follows from not being image-closed characteristic. |
verbal subgroup | generated by set of words | No | Follows from not being image-closed fully invariant. |
GAP implementation
The group and subgroup pair can be constructed using GAP as follows:
G := SmallGroup(16,8); H := Filtered(Subgroups(G), x -> Order(x) = 8 and IsDihedralGroup(x))[1];
The GAP display is as follows:
gap> G := SmallGroup(16,8); H := Filtered(Subgroups(G), x -> Order(x) = 8 and IsDihedralGroup(x))[1]; <pc group of size 16 with 4 generators> Group([ f4, f3, f2 ])
Here is GAP code to verify some of the assertions on this page:
gap> Order(G); 16 gap> Order(H); 8 gap> Index(G,H); 2 gap> StructureDescription(H); "D8" gap> StructureDescription(G/H); "C2" gap> IsNormal(G,H); true gap> IsCharacteristicSubgroup(G,H); true gap> IsFullinvariant(G,H); true