What's the use of composition series in group theory?
I'm going through Dummit & Foote, and I stumbled upon a definition of composition series (for groups) and Jordan-Holder theorem. I get the ''it's something like a factoring of a group'' intuition that authors try to give, but that's a little bit too vague for me. There are no exercises (after that chapter) where we use composition series to prove something interesting about groups (that kind of exercises usually illuminates definitions that seem unnecessary at first).
Factoring of an integer can give us a lot of information about that number, but I can't really see what kind of information composition series and composition factors give us...
Can someone make this clearer for me? Give more motivation to this definition or some examples of usage of composition series?
Solution 1:
There are two basic reasons:
- They give new invariants.
- They let you do proofs by induction.
The purpose of the Jordan-Holder theorem is (1), the length of a composition series and the isomorphism classes that show up in it are uniquely associated to the group, they are not an artifact of the specific choice of a composition series (as there can be more than one). So if you need to know whether two groups are isomorphic and you can calculate this information and see that it's different, then you can conclude that the two groups are not isomorphic.
For the second it has already been suggested that you look up solvable groups, which are defined by the type of composition series they have. These show up, as Jyrki noted, in Galois theory. Also check out the last proof in these notes for an easier example that uses solvable groups.