Composition Series/Examples/Symmetric Group S4

Example of Composition Series

There is $1$ composition series of the symmetric group on $4$ letters $S_4$, up to isomorphism:

$\set e \lhd C_2 \lhd K_4 \lhd A_4 \lhd S_4$

where:

$A_4$ is the alternating group on $4$ letters

$K_4$ is the Klein four-group

$C_n$ is the cyclic group of order $n$

Hence $S_4$ is solvable.

Sources

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Symmetric Groups: $\S 84 \alpha$