Center of dihedral group: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) cyclic group:Z2 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 dihedral group:D8.
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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 subgroup of interest is the subgroup .
The quotient group is isomorphic to dihedral group:D8.
The subgroup is isomorphic to cyclic group:Z2 and the multiplication table is given below. Note that the group is an abelian group, so we do not need to worry about left/right issues:
Element/element | ||
---|---|---|
Contents
Cosets
The subgroup has order 2 and index 8, so it has 8 left cosets. It is a normal subgroup, so the left cosets coincide with the right cosets. The cosets are:
The quotient group is isomorphic to dihedral group:D8, and the multiplication table on cosets is given below. The row element is multiplied on the left and the column element is multiplied on the right.
Element/element | ||||||||
---|---|---|---|---|---|---|---|---|
Arithmetic functions
Function | Value | Explanation |
---|---|---|
order of whole group | 16 | |
order of subgroup | 2 | |
index of subgroup | 8 | |
size of conjugacy class (=index of normalizer) | 1 | |
number of conjugacy classes in automorphism class | 1 |
Subgroup-defining functions
Subgroup-defining function | What it means in general | Why it takes this value | GAP verification (set G := DihedralGroup(16); H := Center(G);) -- see more at #GAP implementation |
---|---|---|---|
center | elements that commute with every group element | No element of the form can be central since it does not commute with . Among powers of , an element is central iff it equals its inverse. The only two such elements are and . | Definitional |
third member of lower central series | commutator subgroup between whole group and its derived subgroup, i.e., | Derived subgroup is , and taking the commutator again gives . | H = CommutatorSubgroup(G,DerivedSubgroup(G)); using CommutatorSubgroup and DerivedSubgroup |
socle | join of all minimal normal subgroups | In fact, it is the unique minimal normal subgroup -- the group is a monolithic group. | H = Socle(G); using Socle |
second agemo subgroup | subgroup generated by elements of the form , here so subgroup generated by all elements of the form | Every fourth power is either or . | H = Agemo(G,2,2); using Agemo |
Subgroup properties
Invariance under automorphisms and endomorphisms
Property | Meaning | Satisfied? | Explanation | GAP verification (set G := DihedralGroup(16);H:= Center(G);) -- see more at #GAP implementation |
---|---|---|---|---|
normal subgroup | invariant under inner automorphisms | Yes | center is normal | IsNormal(G,H); using IsNormal |
characteristic subgroup | invariant under all automorphisms | Yes | center is characteristic | IsCharacteristicSubgroup(G,H); using IsCharacteristicSubgroup |
fully invariant subgroup | invariant under all endomorphisms | Yes | lower central series members are fully invariant, agemo subgroups are fully invariant | IsFullinvariant(G,H); using IsFullinvariant |
verbal subgroup | generated by set of words | Yes | Lower central series members are verbal, agemo subgroups are verbal | ? |
normal-isomorph-free subgroup | no other isomorphic normal subgroup | Yes | ? | |
isomorph-free subgroup, isomorph-containing subgroup | No other isomorphic subgroups | No | There are other subgroups of order two, such as . | |
isomorph-normal subgroup | Every isomorphic subgroup is normal | No | There are other subgroups of order two that are not normal: etc. | |
homomorph-containing subgroup | contains all homomorphic images | No | There are other subgroups of order two. | |
1-endomorphism-invariant subgroup | invariant under all 1-endomorphisms of the group | Yes | It is precisely the set of fourth powers, which must therefore go to squares under 1-endomorphisms | |
1-automorphism-invariant subgroup | invariant under all 1-automorphisms of the group | Yes | Follows from being 1-endomorphism-invariant. | |
quasiautomorphism-invariant subgroup | invariant under all quasiautomorphisms | Yes | Follows from being 1-automorphism-invariant |
GAP implementation
The group and subgroup can be constructed using GAP's DihedralGroup and Center functions as follows:
G := DihedralGroup(16); H := Center(G);
The GAP display looks as follows:
gap> G := DihedralGroup(16); H := Center(G); <pc group of size 16 with 4 generators> Group([ f4 ])
Here is a GAP implementation to verify some of the assertions made on this page:
gap> Order(G); 16 gap> Order(H); 2 gap> Index(G,H); 8 gap> StructureDescription(G/H); "D8" gap> H = CommutatorSubgroup(G,DerivedSubgroup(G)); true gap> H = Socle(G); true gap> H = Agemo(G,2,2); true gap> IsNormal(G,H); true gap> IsCharacteristicSubgroup(G,H); true gap> IsFullinvariant(G,H); true