Center of quaternion group

Definition
This article is about the quaternion group:

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

with multiplication table:

We are interested in the subgroup:

$$H = \{ 1, -1 \}$$

with multiplication table:

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:

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.