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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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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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