Permutation groups generated by short elements
نویسنده
چکیده
This document was provoked by a question on the SEQFAN mailing list, which was passed on to me by Simone Severini. We consider the following four questions. How many subgroups of the symmetric group Sn are generated by • transpositions; • 3-cycles; • transpositions and 3-cycles, • products of pairs of transpositions? For each of the four questions there are three ways of counting the subgroups: • the total number; • the number up to conjugacy in Sn; • the number up to isomorphism. We solve these problems completely for the first three types of generation (noting incidentally that the numbers of isomorphism and conjugacy classes are the same). The fourth is rather more complicated, and all three counts give different values; but the numbers can in principle be calculated from the analysis given here.
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