“Everything should be made as simple as possible, but not simpler.”
―Albert Einstein
SubsectionComposition Series
Definition6.30.
A normal series is called a composition series if each subqoutient \(N_i/N_{i-1}\) is a simple group.
Example6.31.
Suppose \(G = \langle x \rangle\) is cyclic of order \(n > 1\) and say \(n = p_1 \cdots p_k\) is a prime factorization of \(n\text{.}\) Then the factors of
\(\{e\} \unlhd\langle r \rangle\unlhd D_{2n}\) is a normal series. Since \(\langle r \rangle\) is cyclic of order \(n\text{,}\) it has a composition series as in the previous example. By inserting this into the length two normal series given here, we obtain a composition series of \(D_{2n}\) with composition factors
Assume \(G\) is a group and \(N \unlhd G\text{.}\) If \(N\) and \(G/N\) both have composition series, then so does \(G\text{.}\) Moreover, in this case, the list of composition factors of \(G\) is the concatenation of the lists of composition factors of \(N\) and \(G/N\text{.}\)
Proof.
This is a consequence of the Isomorphism Theorems. In detail, a composition series for \(G/N\) has the form
where \(N \unlhd M_1 \unlhd\cdots \unlhd M_k = G\text{.}\) If \(e \unlhd N_1 \unlhd\cdots \unlhd N_j = N\) is a composition seires for \(N\text{,}\) then
\begin{equation*}
e \unlhd N_1 \unlhd\cdots \unlhd N_j \unlhd M_1 \unlhd\cdots \unlhd M_k = G
\end{equation*}
is a composition series for \(G\) since \(M_i/M_{i-1} \cong (M_i/N)/(M_{i-1}/N)\) for all \(i = 1, \dots, k\text{.}\) (For the case \(i = 1\text{,}\) interpret \(M_0\) as \(N\text{.}\))
Example6.34.
As you showed in the homework, \(S_4\) has a normal subgroup \(V\) of order \(4\text{.}\)\(V\) has a composition series of the form
and so the composition factors of \(V\) are \(\mathbb{Z}/2, \mathbb{Z}/2\text{.}\)
The group \(S_4/V\) has order \(6\) and it has an element of order \(3\text{,}\) namely \((1 2 3)\text{.}\) If \(G\) is any group of oder \(6\) with an element \(x\) or order \(3\text{,}\) then the composition factors of \(G\) are \(\mathbb{Z}/2, \mathbb{Z}/3\text{.}\) (In fact, every group of order \(6\) has an element of order \(3\text{.}\)) To see this, note that \(\langle x \rangle\) is normal in \(G\) since it has index two.
Using the Proposition, the compition factors of \(S_4\) are
We can make this more explicit too. The subgroup of \(G/N\) generated by \((1 \, 2 \, 3) N\) corresponds to the subgroup \(M\) of \(G\) that is generated by the elements of \(V\) along with \((1 \, 2 \, 3)\text{.}\) We have
\begin{equation*}
\{e\} \unlhd\langle (1 2)(3 4) \rangle \unlhd V \unlhd M \unlhd G
\end{equation*}
with factors \(\mathbb{Z}/2, \mathbb{Z}/2, \mathbb{Z}/3, \mathbb{Z}/2\text{.}\)
Example6.35.
The composition factors of \(S_5\) are \(\mathbb{Z}/2\) and \(A_5\text{.}\) This holds since \(A_5 \unlhd S_5\text{,}\)\([S_5:A_5] = 2\text{,}\) and \(A_5\) is simple, facts that we will prove later.
We close this subsection with some big-picture type comments. A normal subgroup of a group is analogous to a factor of an integer, and a composition series of a finite group is analogous to a prime factorizations of integers. In this analogy, the role of a prime number is played by a simple group. One major difference is that, unlike for prime factorizations, the list of composition factors of a group do not uniquely determine the group since, for example, \(D_{2n}\) and \(\mathbb{Z}/(2n)\) (and, for that matter, any solvable group of order \(2n\)) have the same composition factors but are certainly not isomorphic.
Nonetheless, a major motivating idea in finite group theory is the following: In order to know everything about finite group theory, one merely needs to know:
a complete list of all simple groups and
a method of reconstructing all possible finite groups having a given list of composition factors.
The former task was essentially accomplished by the 1980’s, when all finite simple groups were classified into a handful of families along with \(26\) “sporadic” simple groups. The latter task is probably not really possible as stated here. But, there is a large collection of properties \(\mathcal P\) of finite groups such that if \(\mathcal P\) holds for all members of a compostion series of a given group then \(\mathcal P\) also holds for the group itself. For such a property \(\mathcal P\text{,}\) it suffices to check that \(\mathcal P\) holds true for every finite simple group. This technique has been used with success.
SubsectionSolvable Groups
Question6.36.The What, Where, and Why of Group Presentations.
Why are solvable groups important? Where are they found? What are they used for?
Answer.
Galois Theory.
A key result of Galois theory is that a polynomial equation is solvable by radicals (i.e., its roots can be expressed using basic arithmetic operations and roots) if and only if its associated Galois group is solvable. This result connects group theory directly to classical problems in algebra.
Lie Groups and Algebraic Groups.
In the study of Lie groups, solvable groups often serve as a starting point for understanding more complex groups. A classic example is the Borel subgroups in the theory of algebraic groups, which are connected solvable groups and play a central role in the classification of algebraic groups.
Representation Theory.
The representations of solvable groups are particularly well-understood due to the fact that these groups have abelian quotients at each level of their derived series. This allows techniques from the representation theory of abelian groups to be extended to solvable groups.
Topology.
In the classification of \(3\)-manifolds, the fundamental group of certain types of 3-manifolds (such as Seifert-fibered spaces) is solvable. Understanding these groups is crucial for classifying the topology of the manifold.
Algebraic Geometry.
The knot group of a knot \(K\) in \(3\)-dimensional space is the fundamental group of the knot complement (the space obtained by removing the knot from \(\R^3\)). This group is often described using a group presentation.
Cryptography.
Cryptographic protocols, such as those based on the discrete logarithm problem or the conjugacy search problem, sometimes use solvable or nearly solvable groups due to their structured nature.
