Center of central product of D8 and Z4
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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The group is central product of D8 and Z4:
The group has 16 elements:
We are interested in the subgroup:
is a normal subgroup of and has four cosets:
The quotient group is isomorphic to a Klein four-group.
|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]|
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|