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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.
Manje Akukho kalkuli yokuxazulula le nto, kodwa amaphuzu ayo anga kalkuli. Zama enye ngezansi, noma ubhale yakho.
Amaphawu asetshenziswa lapha
Chofoza noma yiluphi uphawu lwesibonakaliso ukuze uthole ukuchaza okuphelele, isithombe, kanye nokuthi iyiphi inhlamvu ekhona ekhuluma ngaso.
Imibuzo abantu bebuza
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.
Amangxenye wale khasi aguqulwe kusuka Judson, Abstract Algebra: Theory and Applications (GFDL 1.3). I-condensed futhi iphinde icaciswe lapha; amaphutha awusizo.
Okuningi Abstract Algebra
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