Groups of order 51 and 39 (Sylow theorems).

[vikkivi]

Homework Statement

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

Homework Equations

Sylow theorems.

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, a^b=a³ >

Are these all of the groups? Am I missing any? Do you think these are sufficient answers?
Thank you!

 

[Mark44]

51 is not prime...

 

[vikkivi]

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!

 

[Dahaka14]

i'm telling tyler