Composition Series/Examples

Examples of Composition Series

Cyclic Group $C_8$

There is $1$ composition series of the cyclic group $C_8$, up to isomorphism:

$\set e \lhd C_2 \lhd C_4 \lhd C_8$

Quaternion Group $Q$

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.

Dihedral Group $D_4$

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.

Dihedral Group $D_6$

There are $3$ composition series of the dihedral group $D_6$, up to isomorphism:

$\set e \lhd C_3 \lhd C_6 \lhd D_6$

$\set e \lhd C_2 \lhd C_6 \lhd D_6$

$\set e \lhd C_3 \lhd D_3 \lhd D_6$

where $C_n$ denotes the cyclic group of order $n$.

Symmetric Group $S_2$

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.

Symmetric Group $S_3$

There is $1$ composition series of the symmetric group on $3$ letters $S_3$, up to isomorphism:

$\set e \lhd A_3 \lhd S_3$

where $A_3$ is the alternating group on $3$ letters.

Hence $S_3$ is (trivially) solvable.

Symmetric Group $S_4$

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.

Symmetric Group $S_n$ for $n > 2$ where $n \ne 4$

Let $n \in \Z$ such that $n > 2$ but $n \ne 4$.

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

$\set e \lhd A_n \lhd S_n$

where $A_n$ is the alternating group on $n$ letters.