Center of quaternion group
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) quaternion group (see subgroup structure of quaternion group).
The subgroup is a normal subgroup and the quotient group is isomorphic to Klein four-group.
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Contents
Definition
This article is about the quaternion group:
with multiplication table:
In the table below, the row element is multiplied on the left and the column element on the right.
Element | ||||||||
---|---|---|---|---|---|---|---|---|
We are interested in the subgroup:
with multiplication table:
Element/element | 1 | -1 |
---|---|---|
1 | 1 | -1 |
-1 | -1 | 1 |
Cosets
The subgroup is a normal subgroup, so its left cosets coincide with its right cosets. There are four cosets, given as follows:
The quotient group is isomorphic to a Klein four-group, and its multiplication table is as given below:
Element/element | ||||
---|---|---|---|---|
Subgroup-defining functions
Subgroup-defining function | What it means in general | Why it takes this value | Corresponding quotient-defining function | GAP verification (set G := SmallGroup(8,4); H := Center(G);) -- see #GAP implementation |
---|---|---|---|---|
center | set of elements that commute with every element | inner automorphism group | Definitional | |
derived subgroup | subgroup generated by commutators of elements | abelianization | H = DerivedSubgroup(G); using DerivedSubgroup | |
Frattini subgroup | intersection of all maximal subgroups | It is the intersection of the three maximal subgroups, which are , , and . | Frattini quotient | H = FrattiniSubgroup(G); using FrattiniSubgroup |
socle | join of all minimal normal subgroups | It is the unique minimal normal subgroup | ? | H = Socle(G); using Socle |
First omega subgroup | subgroup generated by elements of order , in this case two | ? | ||
First agemo subgroup | subgroup generated by powers, in this case squares | ? | H = Agemo(G,2,1); using Agemo |
Subgroup properties
Invariance under automorphisms and endomorphisms
Property | Meaning | Satisfied? | Explanation |
---|---|---|---|
normal subgroup | invariant under inner automorphisms | Yes | center is normal, derived subgroup is normal |
characteristic subgroup | invariant under all automorphisms | Yes | center is characteristic, derived subgroup is characteristic |
fully invariant subgroup | invariant under all endomorphisms | Yes | derived subgroup is fully invariant |
isomorph-free subgroup, isomorph-containing subgroup | no other isomorphic subgroups | Yes | only subgroup of order two |
homomorph-containing subgroup | contains every homomorphic image | Yes | on account of its non-identity elements being precisely the elements that are of order two. |
variety-containing subgroup | Yes | ||
verbal subgroup | generated by set of words | Yes | derived subgroup is verbal, agemo subgroups are verbal |
Property | Meaning | Satisfied? | Explanation |
---|---|---|---|
central subgroup | contained in the center | Yes | |
central factor | product with centralizer is whole group | 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:Z4).
For more, see second cohomology group for trivial group action of V4 on Z4.