Center of dihedral group:D8
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:D8 (see subgroup structure of dihedral group:D8).
The subgroup is a normal subgroup and the quotient group is isomorphic to Klein four-group.
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This article discuss the dihedral group of order eight and its center, which is a cyclic group of order two.
The dihedral group of order eight is defined as:
.
It has multiplication table:
The row element is multiplied on the left and the column element is multiplied on the right.
Element | ||||||||
---|---|---|---|---|---|---|---|---|
and the center is the cyclic subgroup:
.
It has multiplication table:
Element/element | ||
---|---|---|
Contents
Cosets
The subgroup has the following four cosets:
The quotient group is isomorphic to Klein four-group, and the multiplication table is as follows:
Element/element | ||||
---|---|---|---|---|
Complements
The subgroup has no permutable complement and also has no lattice complement.
Property | Meaning | Satisfied? | Explanation |
---|---|---|---|
complemented normal subgroup | normal subgroup with a complement. | No | nilpotent and non-abelian implies center is not complemented |
permutably complemented subgroup | subgroup with a permutable complement. | No | nilpotent and non-abelian implies center is not complemented |
lattice-complemented subgroup | subgroup with a lattice complement. | No | nilpotent and non-abelian implies center is not complemented |
Arithmetic functions
Function | Value | Explanation |
---|---|---|
order of whole group | 8 | |
order of subgroup | 2 | |
index of subgroup | 4 | |
size of conjugacy class of subgroup (=index of normalizer) | 1 | center is normal, so the conjugacy class has size 1 |
number of conjugacy classes in automorphism class of subgroup | 1 | center is characteristic |
size of automorphism class of subgroup | 1 | center is characteristic |
Effect of subgroup operators
Function | Value as subgroup (descriptive) | Value as subgroup (link) | Value as group |
---|---|---|---|
normalizer | the whole group | -- | dihedral group:D8 |
centralizer | the whole group | current page | dihedral group:D8 |
normal core | the subgroup itself | current page | cyclic group:Z2 |
normal closure | the subgroup itself | current page | cyclic group:Z2 |
characteristic core | the subgroup itself | current page | cyclic group:Z2 |
characteristic closure | the subgroup itself | current page | cyclic group:Z2 |
commutator with whole group | the trivial subgroup | current page | trivial group |
Subgroup-defining functions
The subgroup is a characteristic subgroup of the whole group and arises as a result of many subgroup-defining functions on the whole group. Some of these are given below.
Subgroup-defining function | Meaning in general | Why it takes this value | Corresponding quotient-defining function | GAP verification (set G := DihedralGroup(8); H := Center(G);) -- see more at #GAP implementation |
---|---|---|---|---|
center | set of elements that commute with every element | To see that every element of is in the center, note that commutes with both and . To see that no other element is in the center, note that and do not commute. | inner automorphism group | Definitional |
derived subgroup | subgroup generated by commutators of all pairs of elements in the group, smallest subgroup with abelian quotient | The quotient (called the abelianization) is , which is isomorphic to Klein four-group. No smaller subgroup works, because the quotient by the trivial subgroup is isomorphic to , which is non-abelian. | abelianization | H = DerivedSubgroup(G); using DerivedSubgroup |
socle | subgroup generated by all the minimal normal subgroups | It is the unique minimal normal subgroup (hence is also a monolith). In general, for a finite -group, the socle is . See socle equals Omega-1 of center for nilpotent p-group. | H = Socle(G); using Socle | |
Frattini subgroup | intersection of all the maximal subgroups | is the intersection of . | Frattini quotient | H = FrattiniSubgroup(G); using FrattiniSubgroup |
Jacobson radical | intersection of all the maximal normal subgroups | For a group of prime power order, this is same as the Frattini subgroup, because nilpotent implies every maximal subgroup is normal. | ||
Baer norm | intersection of normalizers of all subgroups | Normalizer of and of is . Normalizer of and of is . All other normalizers are the whole group. Intersection of the two proper normalizers is . | ||
first agemo subgroup | subgroup generated by all powers, where is the underlying prime (in this case 2) | . In other words, it is the subgroup generated by the squares. In this case, it is precisely' the set of squares. | H = Agemo(G,2,1); using Agemo | |
ZJ-subgroup | center of the join of abelian subgroups of maximum order | The join of abelian subgroups of maximum order (the Thompson subgroup) is the whole group dihedral group:D8, so its center is . | ||
D*-subgroup | abelian and any element acting quadratically on it acts linearly on it (roughly speaking) | In a group of nilpotency class two, this subgroup coincides with the center |
Subgroup properties
Invariance under automorphisms and endomorphisms
Property | Meaning | Satisfied? | Explanation | GAP verification (set G := DihedralGroup(8); H := Center(G);) -- see #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, derived subgroup is characteristic | IsCharacteristicSubgroup(G,H); using IsCharacteristicSubgroup |
fully invariant subgroup | invariant under all endomorphisms | Yes | derived subgroup is fully invariant, agemo subgroups are fully invariant | IsFullinvariant(G,H); using IsFullinvariant |
verbal subgroup | generated by set of words | Yes | derived subgroup is verbal, agemo subgroups are verbal | |
marginal subgroup | Yes | center is marginal | ||
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. | |
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 squares, 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 |
Property | Meaning | Satisfied? | Explanation |
---|---|---|---|
central subgroup | contained in the center | Yes | In fact, it is equal to the center. |
central factor | Yes | (because it is central). | |
transitively normal subgroup | Yes | (because it is a central factor). | |
SCAB-subgroup | Yes |
Cohomology interpretation
We can think of as an extension with abelian normal subgroup and quotient group . Since is in fact the center, the action of the quotient group on the normal subgroup is the trivial group action. We can thus study as an extension group arising from a cohomology class for the trivial group action of (which is a Klein four-group) on (which is cyclic group:Z2).
For more, see second cohomology group for trivial group action of V4 on Z2.
GAP implementation
The group and subgroup pair can be defined using GAP's DihedralGroup and Center functions as follows:
G := DihedralGroup(8); H := Center(G);
The GAP display looks as follows:
gap> G := DihedralGroup(8); H := Center(G); <pc group of size 8 with 3 generators> Group([ f3 ])