[Math] What’s the use of composition series in group theory

abstract-algebragroup-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?

Best Answer

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.

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[Math] the importance of examples in the study of group theory

There is some pedagogical value to understanding how theorems/definitions apply to and playout with particular groups (i.e., through examples). While it is admirable to yearn for a thorough theoretical understanding of group theory, try to think of studying and probing the examples as a means of testing your understanding of the theories and definitions, as they relate to specific groups. Being able to construct a proof is wonderful, but being able to use and apply your learning is crucial to obtaining a fluency with what you're learning. Being able to understand and apply the theorems and definitions is also a test of your comprehension, and will facilitate the proof-writing process as well. For example, being able to construct and anticipate counterexamples is crucial to proof-writing, and counterexamples will be hard to come by if you don't understand how particular groups exemplify or fail to exemplify group properties.

A great website is the website Groupprops - "A Group Properties Wiki". You'll find a menu to the left:

Terms/definitions
Facts/theorems
Survey articles
Specific information

and can quickly look-up (almost) anything/everything you'd want to know about (almost) any group.

[Math] The Jordan-Hölder Theorem

If $N_{r-1}=M_{s-1}$ then you can use induction, so assume not.

Choose a composition series of $H$. Now we have four composition series of $G$, your two original ones, and two that go through $H$ (it's best to draw a picture). Two of the series go through $N_{r-1}$ and the other two go through $M_{s-1}$, so you can apply induction to both of those.

To finish off the proof, you need the isomorphisms $G/N_{r-1} \cong M_{s-1}/H$ and $G/M_{s-1} \cong N_{r-1}/H$, which come from the Second Isomorphism Theorem.

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[Math] the importance of examples in the study of group theory

[Math] The Jordan-Hölder Theorem

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