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Math Help - Sylow-p subgroups of A4

  1. #1
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    Sylow-p subgroups of A4

    12=2^2*3..So i will have to list the Sylow-2 subgroup of A4 whose cardinality will be 2^2=4 and Sylow-3 subgroup whose cardinality will be 3.

    one of the book that i refered said that there are 4 Sylow-3 subgroups of cardinality 3 and 1 Sylow-2 subgroup of cardinailty 4..

    Can anyone please explain me this!
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  2. #2
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    I got it!i need to use the fact that if np is the number of Sylow-p subgroup of group G then np divides the index if P in G,(where Pis the Sylow-p subgroup of G)..
    So if P1 is the Sylow-2 subgroup of A4 then np is either 1 or 3,where cardinality of P1 is 4..so there is only one Sylow-2 subgroup of A4 as there are only three elements in A4 of order 2..(so these three elements along with identity form the only Sylow-2 subgroup of A4)..
    Similarly for Sylow-3 subgroup of A4..The possibilty of np is 1or 2 or 4..nand then as above..

    Correct??
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