Composition Series/Examples/Quaternion Group Q

From ProofWiki
Jump to navigation Jump to search

Example of Composition Series

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.


Proof

Let $Q$ be defined as by Group Presentation of Quaternion Group:

$Q = \gen {\alpha, \beta: \alpha^4 = e, \beta^2 = \alpha^2, \alpha \beta \alpha = \beta}$


From Subgroups of Quaternion Group, $Q$ has $3$ subgroups of index $2$:

\(\ds \) \(\) \(\ds \set {e, \alpha, \alpha^2, \alpha^3}\)
\(\ds \) \(\) \(\ds \set {e, \beta, \alpha^2, \alpha^2 \beta}\)
\(\ds \) \(\) \(\ds \set {e, \alpha \beta, \alpha^2, \alpha^3 \beta}\)

We have that:

$\set {e, \alpha, \alpha^2, \alpha^3} = \gen \alpha = C_4$

and:

$\set {e, \alpha^2, \beta, \beta \alpha^2} = D_2 = K_4$

and:

$\set {e, \alpha \beta, \alpha^2, \alpha^3 \beta} = D_2 = K_4$


By Subgroup of Index 2 is Normal, these are normal.

Thus we have so far:

$\set e \lhd \cdots \lhd C_4 \lhd Q$
$\set e \lhd \cdots \lhd K_4 \lhd Q$


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:

$\set e \lhd C_2 \lhd K_4 \lhd Q$

$\blacksquare$


Sources