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Math Help - determine up to isomorphism all groups of order 13^2*3^2

  1. #1
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    determine up to isomorphism all groups of order 13^2*3^2

    Here's my problem:

    determine up to isomorphism all groups of order 13^2*3^2

    so far I can do this

    Let G be a group such that |G| = 13^2 * 3^2

    n_3 = # of 3-sylow subgroups of G
    n_13 = # of 13-sylow subgroups of G

    n_3 = 1 (mod 3) and n_3 divides 13^2, thus n_3 = 1, 13 or 169
    n_13 = 1 (mod 13) and n_13 divides 3^2, thus n_13 = 1

    since the 13-sylow subgroup is unique, it must be normal in G

    and here's where I go all fuzzy. Any help is appreciated.

    I also found this thread: http://www.mathhelpforum.com/math-he...omorphism.html

    which describes a similar problem, but wasn't sure if there would be further considerations since both primes are squared.
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  2. #2
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    We have  |G| = 3^2 13^2 = 1521 .

    Since the order of G has 2 distinct prime factors, 3 and 13, Sylow Theorem I (henceforth to be referred to as Sylow I) tells us that G has at least 1 Sylow-3 subgroup and at least one 1 Sylow-13 subgroup. Let's find out how many of each.

    We will denote the number of Sylow-3 subgroups and Sylow-13 subgroups by  n_3 \text{  and  } n_{13}  , respectively.

    Sylow III tells us that the number of subgroups of G of order  3^2 is congruent to 1 modulo 3 and is a factor of  \frac{|G|}{3^2} = 169 . Likewise, the number of subgroups of order  13^2 is congruent to 1 modulo 13 and is a factor of  \frac{|G|}{13^2} = 9 . Moreover, since a subgroup of a prime power order is a Sylow-p subgroup (and thus is isomorphic to every other Sylow-p subgroup), we have


     n_3 \equiv 1 \pmod3 \text{   and  } n_3 | 169  <br />
\implies n_3 = 1, 13, \text{ or } 169

     n_{13} \equiv 1 \pmod3 \text{   and  } n_{13} | 9 <br />
\implies n_{13} = 1


    There are 13 isomorphism types of finite groups of order 1521. They are:

    C1521
    C169 : C9
    C39 x C39
    C507 x C3
    C117 x C13
    C3 x (C169 : C3)
    C3 x ((C13 x C13) : C3)
    C3 x ((C13 x C13) : C3)
    C13 x (C13 : C9)
    (C13 x C13) : C9
    (C13 x C13) : C9
    (C13 : C3) x (C13 : C3)
    C39 x (C13 : C3)
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