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By a Classification of Dickson everysubgroup of PSL(2,p) has index at least p+1

is there an easy proof with out this classification??

What can be said about the minimal index of subgroups PSL(r,q)?? There is a classification of subgroups of PSL(3,p) by Bloom , even for this list i could not calculate all the indexes.

Any references are welcome

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1 Answer 1

up vote 5 down vote accepted

The first published proof that the index of a subgroup of PSL$(2,p)$ is at least $p+1$ for primes $p \ge 13$ is in:

C. Jordan, "Note sur les equations modulaires", C.R. Acad. Sci. Paris 66 (1868), 308-312,

a long time before the classification!

The minimal indexes of subgroups of classical simple groups are determined (again pre-classification) in

B.N. Cooperstein, "Minimal degree for a permutation representation of a classical group", Israel J. Math. 30 (1978), 213-225.

There are apparently a couple of mistakes in Cooperstein's paper, but they concern $U_n(2)$ and orthogonal groups over $F_3$.

In particular, the minimal index of PSL$(n,q)$ is $(q^n-1)/(q-1)$ except for $(n,q)$ = (2,5), (2,7), (2,9), (2,11), or (4,2).

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@Derek: This is of course known. The challenge, as I see it, is to give an elementary proof for $PSL(2,p)$. – Mark Sapir Nov 4 '10 at 13:35
@Derek: I do not think that by "classification" chemaida meant classification of finite simple groups; I think he meant the classification of subgroups of $PSL(2,p)$. – Mark Sapir Nov 4 '10 at 13:46
Thank you all for the answers, Derek in particular managed to give precise references for exactlay what i was looking for. The comment about p > 11 is also correct it supposedly goes back to Everiste Galois – chemaida Nov 4 '10 at 14:02
@Mark: Yes I am sure you are right about what was meant by "classification"! But Jordan's paper predates Dickson's classification of subgroups of PSL$(2,q)$. – Derek Holt Nov 4 '10 at 14:28

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