Composition Series/Examples/Quaternion Group Q

Example of Composition Series
There are $2$ composition series of the quaternion group $Q$, up to isomorphism:
 * $\set e \lhd C_2 \lhd C_4 \lhd Q$


 * $\set e \lhd C_2 \lhd K_4 \lhd Q$

where:
 * $C_n$ denotes the cyclic group of order $n$.
 * $K_4$ denotes the Kline $4$-group.

Proof
Let $Q$ be defined as by Group Presentation of Quaternion Group:


 * $Q = \gen {\alpha, \beta: \alpha^4 = e, \beta^2 = \alpha^2, \alpha \beta \alpha = \beta}$

From Subgroups of Quaternion Group, $Q$ has $3$ subgroups of index $2$:

We have that:
 * $\set {e, \alpha, \alpha^2, \alpha^3} = \gen \alpha = C_4$

and:
 * $\set {e, \alpha^2, \beta, \beta \alpha^2} = D_2 = K_4$

and:
 * $\set {e, \alpha \beta, \alpha^2, \alpha^3 \beta} = D_2 = K_4$

By Subgroup of Index 2 is Normal, these are normal.

Thus we have so far:


 * $\set e \lhd \cdots \lhd C_4 \lhd Q$


 * $\set e \lhd \cdots \lhd K_4 \lhd Q$

We have from Cyclic Group is Abelian and Subgroup of Abelian Group is Normal that all subgroups of $C_4$ are normal in $C_4$.

This leads directly to the composition series:


 * $\set e \lhd C_2 \lhd C_4 \lhd D_4$

Similarly, the Kline $4$-group is abelian, and it has subgroups of order $2$, that is, $C_2$.

This leads to the composition series:


 * $\set e \lhd C_2 \lhd K_4 \lhd Q$