Quaternion group

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Definition

The quaternion group is a group with eight elements, which can be described in any of the following ways:

• It is the holomorph of the ring $\mathbb {Z} /4\mathbb {Z}$.
• It is the holomorph of the cyclic group of order 4.
• It is the group comprising eight elements $1,-1,i,-i,j,-j,k,-k$ where 1 is the identity element, $(-1)^{2}=1$ and all the other elements are squareroots of $-1$, such that $(-1)i=-i,(-1)j=-j,(-1)k=-k$ and further, $ij=k,ji=-k,jk=i,kj=-1,ki=jik=-j$ (the remaining relations can be deduced from these).

Group properties

The quaternion group is an example of a group in which every subgroup is normal, even though the group is not Abelian. In other words, it is a Dedekind group.

Nilpotence

This particular group is nilpotent

Solvability

This particular group is solvable

Abelianness

This particular group is not Abelian

Simplicity

This particular group is not simple

Generalization to other primes

The construction of the quaternion group can be mimicked for other primes giving, in general, a non-Abelian group of order $p^{3}$. The general construction involves taking a semidirect product of the cyclic group of order $p^{2}$ with a subgroup of order $p$ in the automorphism group, say the subgroup generated by the automorphism taking an element to its $(p+1)^{th}$.

Subgroup-defining functions

Center

The center of this group is abstractly isomorphic to: [[{{{1}}}]]Property "Center" (as page type) with input value "{{{1}}}" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.

The center of the quaternion group is the two-element subgroup comprising $-1$ and $1$.

Commutator subgroup

The commutator subgroup of this group is abstractly isomorphic to: [[{{{1}}}]]Property "Commutator subgroup" (as page type) with input value "{{{1}}}" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.

The commutator subgroup of the quaternion group is the same as its center: the two-element subgroup comprising $-1$ and $1$.

In particular this shows that the quaternion group is a group of nilpotence class two.

{frattini subgroup}}

The Frattini subgroup is also the same as the center and commutator subgroup. In fact, this makes the quaternion group into an extraspecial group.

Subgroups

Normal subgroups

All subgroups are normal. The subgroups are the whole group, the trivial subgroup, the center, and three copies of the cyclic group on 4 elements.

Characteristic subgroups

There are only three characteristic subgroups: the whole group, the trivial subgroup and the center.

Fully characteristic subgroups

The fully characteristic subgroups of this group are the same as its characteristic subgroups.