Composition Series/Examples/Symmetric Group S2

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: $\S 84 \alpha$