Center of direct product of D8 and Z2
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) Klein four-group and the group is (up to isomorphism) direct product of D8 and Z2 (see subgroup structure of direct product of D8 and Z2).
The subgroup is a normal subgroup and the quotient group is isomorphic to Klein four-group.
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Contents
Definition
The group is direct product of D8 and Z2, given as follows:
The subgroup is the first direct factor (dihedral group:D8) and the subgroup is the second direct factor (cyclic group:Z2).
has 16 elements:
The subgroup we are interested in is:
is the center of and is isomorphic to the Klein four-group.
Cosets
is a normal subgroup of and it has four cosets:
The quotient group is also isomorphic to the Klein four-group.
Arithmetic functions
Function | Value | Explanation |
---|---|---|
order of whole group | 16 | |
order of subgroup | 4 | |
index of subgroup | 4 | |
size of conjugacy class of subgroup | 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 | GAP verification (set G := DirectProduct(DihedralGroup(8),CyclicGroup(2));H := Center(G)) |
---|---|---|---|
center | precisely the elements that commute with every element | center of direct product is direct product of centers, and the center of dihedral group:D8 is | H = Center(G); using Center |
socle | join of all minimal normal subgroups. For a group of prime power order, same as (because minimal normal implies central in nilpotent). | Center is already elementary abelian, so it equals the socle. | H = Center(G); using Center |
Subgroup properties
Invariance under automorphisms and endomorphisms: basic properties
Property | Satisfied? | Explanation | GAP verification (set G := DirectProduct(DihedralGroup(8),CyclicGroup(2));H := Center(G)) -- see #GAP implementation |
---|---|---|---|
normal subgroup | Yes | center is normal | IsNormal(G,H); using IsNormal |
characteristic subgroup | Yes | center is characteristic | IsCharacteristicSubgroup(G,H); using IsCharacteristicSubgroup |
fully invariant subgroup | No | Consider the endomorphism . This endomorphism does not preserve the subgroup. | IsFullinvariant(G,H); using IsFullinvariant |
Property | Meaning in general | Satisfied? | Explanation | GAP verification (set G := DirectProduct(DihedralGroup(8),CyclicGroup(2));H := Center(G)) -- see #GAP implementation |
---|---|---|---|---|
marginal subgroup | (complicated) | Yes | center is marginal | |
verbal subgroup | generated by set of words | No | verbal implies fully invariant, the subgroup is not fully invariant | |
finite direct power-closed characteristic subgroup | any finite direct power is characteristic in the corresponding direct power of the whole group | Yes | center is finite direct power-closed characteristic (also via marginal) |