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Groups of order 51 and 39 (Sylow theorems).

  1. Dec 9, 2009 #1
    1. The problem statement, all variables and given/known data

    a) Classify all groups of order 51.
    b) Classify all groups of order 39.

    2. Relevant equations
    Sylow theorems.
    3. The attempt at a solution

    a) C51
    b) Z3 X Z13
    and Z13 x Z3, the semi-direct product with presentation <a,b|a13=b3=1, ab=a3 >

    Are these all of the groups? Am I missing any? Do you think these are sufficient answers?
    Thank you!
  2. jcsd
  3. Dec 9, 2009 #2


    Staff: Mentor

    51 is not prime...
  4. Dec 13, 2009 #3
    Yes, I understand that 51 is not a prime, but 51 is a multiple of 3 and 17.

    So, I'm using this theorem:

    Theorem 1. Suppose G is a non-Abelian group whose order is divisible by at least two
    distinct primes and all of whose proper subgroups have prime-power order. Then
    (i) absolutevalue[G] = p" q where p and q are primes;
    (ii) the Sylow p-subgroup of G is the unique nontrivial proper normal subgroup of G and
    is elementary Abelian;
    (iii) absolutevalue[G'] = p";
    (iv) absolutevalue[Z(G)] = 1;
    (v) G has p" Sylow q-subgroups and when n > 1, q divides (p" - t)/(p - 1).

    I know that groups of order 3 are the C3.
    Groups of order 17 are C17.
    So, I think that 51 must be C51.
    Any help would be appreciated!
    Last edited: Dec 13, 2009
  5. Dec 16, 2009 #4
    i'm telling tyler
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