Actually, you want to avoid these fuller finite matrix groups; For instance, a lot of the PSL(n,q) groups are simple! (Exceptions can occur for small n, and small finite fields q.)

Note that the group PSL(n,q) is just the quotient of SL(n,q) by it center (the diagonal matrices in SL(n,q)), so therefore the groups SL(n,q) are not far off from being simple.

Bogopolsky's book does a very nice job explaining in general terms how simplicity arises, and here are some hallmarks:

1) your group is highly transitive (say, at least two-transitive, for example) in its action of the vector space of the appropriate size,

2) your group has many elements of ``small support'' (given a point in the space you are acting on, there are elements in the group which only move this point and a few others).

3) your group is equal to its commutator subgroup.

In fact, these hallmarks were recognised by Higman in a paper going back to the 1950's, and then again in a well written short paper by D.B.A. Epstein (``The simplicity of certain groups of homeomorphisms'' Compositio Mathematica, 1970, pp 165--173), where in both cases, these hallmarks were looked for in infinite groups.

So, as any finite group can be found as a subgroup of permutation groups, and permutation groups are linear (embed any permutation group in a group of permutations of the major axes, which is a linear group). We see that there are lots of linear groups with lots of normal subgroups; just not really the ones you mentioned.

Dear Collin,

thanks for the informative answer, but instead of telling what I should not do, maybe you can tell me what I should do? ;=). Actually one of my motivations is to construct permutation groups with many normal subgroups. Maybe a larger question would be :

what is the largest number N(v) of normal subgroups a finite group of given order v might have?

Dear Patrick,

You are welcome.

I suppose the point I was trying to make is that you can model ALL finite groups as matrix groups (all groups are permutation groups), so the question is what properties of groups are likely to produce lots of normal subgroups?

Now for normality, abelian groups work nicely; all subgroups are normal! BUT, you want to have lots of different quotients. This means you would like to have good ``divisibility properties'' between the orders of the relative elements. This suggests p-groups.

At this point I am amending my old answer; I have discussed this with Dr. MArtyn Quick, and we together came to undersatnd the (I believe) correct answer. Here is the meat of that discussion.

Now, for a p-group, the ones with the most normal subgroups are again abelian, and as (for instance) the number of subgroups of (Z/pZ)^k is exactly the number of linear subspaces of this vector space (as opposed to Z/(p^kZ), which has a unique subgroup for each divisor of p^k), we see this is the local max.

So, in general, the value of N(v) where v = p1^{k1}p2^{k2}...pm^{km} (pi, pj are distinct primes if i< j, and assume pi