Composition Series/Examples/Dihedral Group D4
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Example of Composition Series
There are $2$ composition series of the dihedral group $D_4$, up to isomorphism:
- $\set e \lhd C_2 \lhd C_4 \lhd D_4$
- $\set e \lhd C_2 \lhd K_4 \lhd D_4$
where:
- $C_n$ denotes the cyclic group of order $n$.
- $K_4$ denotes the Kline $4$-group.
Proof
Let $D_4$ be defined as by Group Presentation of Dihedral Group:
- $D_4 = \gen {\alpha, \beta: \alpha^4 = \beta^2 = e, \beta \alpha \beta = \alpha^{−1} }$
From Subgroups of Dihedral Group $D_4$, $D_4$ has $2$ subgroups of index $2$:
- $\set {e, \alpha, \alpha^2, \alpha^3}$
- $\set {e, \alpha^2, \beta, \beta \alpha^2}$
We have that:
- $\set {e, \alpha, \alpha^2, \alpha^3} = \gen \alpha = C_4$
and:
- $\set {e, \alpha^2, \beta, \beta \alpha^2} = D_2 = K_4$
that is, the Kline $4$-group.
By Subgroup of Index 2 is Normal, both of these are normal.
Thus we have so far:
- $\set e \lhd \cdots \lhd C_4 \lhd D_4$
- $\set e \lhd \cdots \lhd K_4 \lhd D_4$
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:
- $\set e \lhd C_2 \lhd K_4 \lhd D_4$
$\blacksquare$
Sources
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Normal and Composition Series: $\S 74 \ \beta$