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When looking at finite group theory research, it seems to me that a lot of energy is devoted to determining the maximal subgroups of certain classes of groups. For example, the O'Nan Scott theorem gives a classification of the maximal subgroups of $S_n$. There is an entire book (link) devoted to studying the maximal subgroups of simple classical groups. I'm pretty sure there are many other examples.

Question: How does understanding the maximal subgroups of a group help us in understanding the structure of the group? Why should we care about maximal subgroups?

This question is of course a bit vague. What I'm looking for is motivation and examples of applications.

For example, I can see how the classification of finite simple groups (CFSG) is useful and interesting in group theory: every group is made up of simple groups (Jordan-Hölder), and CFSG is a very powerful tool for proving things. Many theorems have been proven with the following strategy:

Step 1: Prove that a minimal counterexample is a finite simple group.

Step 2: Check the theorem for each family of finite simple groups.

So the importance of CFSG is clear. What I don't understand currently is why a group theorist would care about maximal subgroups or classifying them.

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I'm not sure you're claim that this is a problem to which "lots of energy" are devoted. When dealing with a certain kind of groups it certainly can be helpful to know something about maximal subgroups of groups (or some of them) of that kind, but it can be helpful as well know something about maximal/minimal abelian/solvable/nilpotent/finite-index/etc. subgroups. –  DonAntonio Feb 11 at 11:31

2 Answers 2

up vote 6 down vote accepted

Here are a couple of fairly trivial observations.

The (conjugacy classes of) maximal subgroups of a group correspond to its primitive permutation representations, and so by understanding them you learn a lot about their potential actions and (at least in the case of simple groups) the smallest degree symmetric groups in which they embed.

If you can compute maximal subgroups of groups, then you can recursively compute all of their subgroups, and some of the implemented algorithms adopt this approach. In any case, the question of whether one given group $H$ embeds in another group $G$ arises very frequently in group theory, and knowing the maximal subgroups of $G$ will certainly help you decide!

Thanks for advertising the "entire book" by the way!

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I have one observation for this: For every element $x$ in a non-cyclic finite group $G$, there exists one maximal subgroup $M$ such that $x \in M$. So if you know the information of all maximal, you will understand the whole group. For example, the finite groups with each maximal subgroups abelian have been classified long time ago, and have many applications.

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