Composition Series/Examples/Symmetric Group S2
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Example of Composition Series
There is $1$ composition series of the symmetric group on $2$ letters $S_2$, up to isomorphism:
- $\set e = A_2 \lhd S_2$
where $A_2$ is the (degenerate) alternating group on $2$ letters.
Hence $S_2$ is (trivially) solvable.
Proof
We have that $S_2$ is isomorphic to the parity group, which is the cyclic group $C_2$.
From Cyclic Group is Abelian and Subgroup of Abelian Group is Normal, all subgroups of $C_n$ are normal in $C_n$.
This leads directly to the composition series:
- $\set e \lhd C_2$
$\blacksquare$
Sources
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Symmetric Groups: $\S 84 \alpha$