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Solvable Groups
A subnormal series of a group G is a finite sequence of subgroups \[\begin{aligned}\end{aligned}\], where H_i is a normal subgroup of H_{i+1}.
Solvable Groups
A subnormal series of a group \(G\) is a finite sequence of subgroups \[\begin{aligned}\end{aligned}\], where \(H_i\) is a normal subgroup of \(H_{i+1}\). If each subgroup \(H_i\) is normal in \(G\), then the series is called a normal series. The length of a subnormal or normal series is the number of proper inclusions.
Example
Any series of subgroups of an abelian group is a normal series. Consider the following series of groups: \[\begin{aligned}{\mathbb Z} \supset 9{\mathbb Z} \supset 45{\mathbb Z} \supset 180{\mathbb Z} \supset \{0\}, \\ {\mathbb Z}_{24} \supset \langle 2 \rangle \supset \langle 6 \rangle \supset \langle 12 \rangle \supset \{0\}\end{aligned}\].
Example
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.
A subnormal (normal) series \(\{ K_j \}\) is a refinement of a subnormal (normal) series \(\{ H_i \}\) if \(\{ H_i \} \subset \{ K_j \}\). That is, each \(H_i\) is one of the \(K_j\).
Example
The series \[\begin{aligned}\end{aligned}\] is a refinement of the series \[\begin{aligned}\end{aligned}\].
The best way to study a subnormal or normal series of subgroups, \(\{ H_i \}\) of \(G\), is actually to study the factor groups \(H_{i+1}/H_i\). We say that two subnormal (normal) series \(\{H_i \}\) and \(\{ K_j \}\) of a group \(G\) are isomorphic if there is a one-to-one correspondence between the collections of factor groups \(\{H_{i+1}/H_i \}\) and \(\{ K_{j+1}/ K_j \}\).
Example
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 \[\begin{aligned}{\mathbb Z}_{60} / \langle 3 \rangle \cong \langle 20 \rangle / \{ 0 \} \cong {\mathbb Z}_{3} \\ \langle 3 \rangle / \langle 15 \rangle \cong \langle 4 \rangle / \langle 20 \rangle \cong {\mathbb Z}_{5} \\ \langle 15 \rangle / \{ 0 \} \cong {\mathbb Z}_{60} / \langle 4 \rangle \cong {\mathbb Z}_4\end{aligned}\].
Example
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.
Condensed: the full section is in Judson, Abstract Algebra: Theory and Applications.
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Spurningar sem fólk spyr
What is a group, in plain words?
A set with one operation that is associative, has an identity, and lets every element be undone. Symmetries of any object form a group. That is where the idea came from.
What is the difference between a ring and a field?
A ring has addition and multiplication that behave like the integers (you cannot always divide); a field is a ring where every non-zero element has a reciprocal, like the rationals or the reals.
Hlutar þessarar síðu eru aðlagað frá Judson, Abstract Algebra: Theory and Applications (GFDL 1.3). Þjappað og endurútskýrt hér; villur eru okkar.
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