Composition Series/Examples/Quaternion Group Q
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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 $2$:
\(\ds \) | \(\) | \(\ds \set {e, \alpha, \alpha^2, \alpha^3}\) | ||||||||||||
\(\ds \) | \(\) | \(\ds \set {e, \beta, \alpha^2, \alpha^2 \beta}\) | ||||||||||||
\(\ds \) | \(\) | \(\ds \set {e, \alpha \beta, \alpha^2, \alpha^3 \beta}\) |
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:
- $\set e \lhd C_2 \lhd K_4 \lhd Q$
$\blacksquare$
Sources
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Normal and Composition Series: $\S 74 \ \beta$