Solvable Groups

Any series of subgroups of an abelian group is a normal series. Consider the following series of groups: \[\begin{aligned}{\mathbb Z} \supset 9{\mathbb Z} \supset 45{\mathbb Z} \supset 180{\mathbb Z} \supset \{0\}, \\ {\mathbb Z}_{24} \supset \langle 2 \rangle \supset \langle 6 \rangle \supset \langle 12 \rangle \supset \{0\}\end{aligned}\].

A subnormal series need not be a normal series. Consider the following subnormal series of the group \(D_4\): \[\begin{aligned}\end{aligned}\]. The subgroup \(\{ (1), (1 \, 2)(3 \, 4) \}\) is not normal in \(D_4\); consequently, this series is not a normal series.

The series \[\begin{aligned}\end{aligned}\] is a refinement of the series \[\begin{aligned}\end{aligned}\].

The two normal series \[\begin{aligned}{\mathbb Z}_{60} \supset \langle 3 \rangle \supset \langle 15 \rangle \supset \{ 0 \} \\ {\mathbb Z}_{60} \supset \langle 4 \rangle \supset \langle 20 \rangle \supset \{ 0 \}\end{aligned}\] of the group \({\mathbb Z}_{60}\) are isomorphic since \[\begin{aligned}{\mathbb Z}_{60} / \langle 3 \rangle \cong \langle 20 \rangle / \{ 0 \} \cong {\mathbb Z}_{3} \\ \langle 3 \rangle / \langle 15 \rangle \cong \langle 4 \rangle / \langle 20 \rangle \cong {\mathbb Z}_{5} \\ \langle 15 \rangle / \{ 0 \} \cong {\mathbb Z}_{60} / \langle 4 \rangle \cong {\mathbb Z}_4\end{aligned}\].

For \(n \geq 5\), the series \[\begin{aligned}\end{aligned}\] is a composition series for \(S_n\) since \(S_n / A_n \cong {\mathbb Z}_2\) and \(A_n\) is simple.