Center of quaternion group

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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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Definition

This article is about the quaternion group:

G := \{ 1, -1, i, -i, j, -j, k,-k \}

with multiplication table:


In the table below, the row element is multiplied on the left and the column element on the right.

Element \! 1 \! -1 \! i \! -i \! j \! -j \! k \! -k
\! 1 \! 1 \! -1 \! i \! -i \! j \! -j \! k \! -k
\! -1 \! -1 \! 1 \! -i \! i \! -j \! j \! -k \! k
\! i \! i \! -i \! -1 \! 1 \! k \! -k \! -j \! j
\! -i \! -i \! i \! 1 \! -1 \! -k \! k \! j \! -j
\! j \! j \! -j \! -k \! k \! -1 \! 1 \! i \! -i
\! -j \! -j \! j \! k \! -k \! 1 \! -1 \! -i \! i
\! k \! k \! -k \! j \! -j \! -i \! i \! -1 \! 1
\! -k \! -k \! k \! -j \! j \! i \! -i \! 1 \! -1


We are interested in the subgroup:

H = \{ 1, -1 \}

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:

\! \{ 1,-1\}, \{ i,-i \}, \{ j,-j \}, \{ k,-k \}

The quotient group is isomorphic to a Klein four-group, and its multiplication table is as given below:

Element/element \! \{ 1, -1 \} \! \{ i, -i \} \! \{ j, -j \} \! \{k -k \}
\! \{ 1, -1 \} \! \{ 1, -1 \} \! \{ i, -i \} \! \{ j, -j \} \! \{ k, -k \}
\! \{ i, -i \} \! \{ i, -i \} \! \{ 1, -1 \} \! \{ k, -k \} \! \{ j, -j \}
\! \{ j, -j \} \! \{ j, -j \} \! \{ k, -k \} \! \{ 1, -1 \} \! \{ i, -i \}
\! \{ k, -k \} \! \{ k, -k \} \! \{ j, -j \} \! \{ i, -i \} \! \{ 1, -1 \}

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 \langle i \rangle, \langle j \rangle, and \langle k \rangle. 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 p, in this case two  ?
First agemo subgroup subgroup generated by p^{th} 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

Centrality and related properties

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 G as an extension with abelian normal subgroup H and quotient group G/H. Since H is in fact the center, the action of the quotient group on the normal subgroup is the trivial group action. We can thus study G as an extension group arising from a cohomology class for the trivial group action of G/H (which is a Klein four-group) on H (which is cyclic group:Z4).

For more, see second cohomology group for trivial group action of V4 on Z4.