Q8 in 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) quaternion group 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 cyclic group:Z2.
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The group is central product of D8 and Z4:
The group is a central product of dihedral group:D8 and cyclic group:Z4, and is also a central product of the quaternion group and cyclic group:Z4. The group has 16 elements:
We are interested in the subgroup:
This group is isomorphic to the quaternion group, with .
Contents
Cosets
The subgroup has index two and index two implies normal, so it is a normal subgroup and thus its left cosets coincide with its right cosets. The two cosets are:
Arithmetic functions
Function | Value | Explanation |
---|---|---|
order of whole group | 16 | |
order of subgroup | 8 | |
index of subgroup | 2 | |
size of conjugacy class of subgroup (= index of normalizer) | 1 | |
number of conjugacy classes in automorphism class of subgroup | 1 | |
size of automorphism class of subgroup | 1 |
Subgroup properties
Invariance under automorphisms and endomorphisms: basic properties
Property | Meaning | Satisfied? | Explanation |
---|---|---|---|
normal subgroup | invariant under all inner automorphisms | Yes | index two implies normal |
characteristic subgroup | invariant under all automorphisms | Yes | Follows from being isomorph-free (see below) |
fully invariant subgroup | invariant under all endomorphisms | No |
Properties based on resemblance and invariance
Property | Meaning | Satisfied? | Explanation |
---|---|---|---|
isomorph-free subgroup | no other isomorphic subgroup | Yes | |
isomorph-containing subgroup | contains every isomorphic subgroup (same as isomorph-free for finite groups) | Yes | |
homomorph-containing subgroup | contains every homomorphic image | No |
Property | Meaning | Satisfied? | Explanation |
---|---|---|---|
cocentral subgroup | product with center of whole group is whole group | Yes | |
central factor | product with centralizer is whole group | Yes | |
central subgroup | contained in the center | No | |
transitively normal subgroup | every normal subgroup of the subgroup is normal in the whole group | Yes | central factor implies transitively normal |
hereditarily normal subgroup | every subgroup of the subgroup is normal in the whole group | Yes | Follows from the subgroup being transitively normal and a Dedekind group -- all its subgroups are normal. See subgroup structure of quaternion group. |