Center of central product of D8 and Z4
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) cyclic group:Z4 and the group is (up to isomorphism) central product of D8 and Z4 (see subgroup structure of central product of D8 and Z4).
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 central product of D8 and Z4:
The group has 16 elements:
We are interested in the subgroup:
This subgroup is isomorphic to cyclic group:Z4 and is the center of .
Cosets
is a normal subgroup of and has four cosets:
The quotient group is isomorphic to a Klein four-group.
Subgroup-defining functions
Subgroup-defining function | Meaning in general | Why it takes this value |
---|---|---|
center | elements that commute with every element | PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] |
Subgroup properties
Invariance under automorphisms and endomorphisms: basic properties
Property | Meaning | Satisfied? | Explanation | GAP verification (set G := SmallGroup(16,13);H := Center(G)) |
---|---|---|---|---|
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 | No | PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] | fully invariant subgroup(G,H); using fully invariant subgroup |