# Groups of order 51 and 39 (Sylow theorems).

• vikkivi
In summary, we are dealing with the task of classifying all groups of order 51 and 39 using the Sylow theorems. After some discussion and application of the theorems, it is determined that the groups of order 51 are C51 and the groups of order 39 are Z3 x Z13 and Z13 x Z3, the semi-direct product with presentation <a,b|a13=b3=1, ab=a3>. It is also noted that these are the only groups and are sufficient answers. The conversation also touches on the fact that 51 is not a prime number and how this is taken into account in the classification process.
vikkivi

## Homework Statement

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

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, ab=a3 >

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

51 is not prime...

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:
i'm telling tyler

## What are Sylow theorems?

Sylow theorems are a set of three theorems in group theory that help in finding the number of subgroups in a finite group. These theorems are named after the mathematician Ludwig Sylow.

## What is the significance of groups of order 51 and 39?

Groups of order 51 and 39 are important because they are two of the smallest non-cyclic groups. They also have interesting properties and can be used to illustrate the concepts of Sylow theorems.

## How do you find the number of Sylow 3-subgroups in a group of order 51?

According to Sylow's first theorem, the number of Sylow 3-subgroups in a group of order 51 must divide 51 and leave a remainder of 1. Therefore, the possible number of Sylow 3-subgroups is 1 or 17. To determine the exact number, we can use Sylow's third theorem which states that the number of Sylow 3-subgroups is congruent to 1 mod 3 and divides 51. Therefore, the only possible number of Sylow 3-subgroups in a group of order 51 is 1.

## Can a group of order 39 have a normal Sylow 13-subgroup?

No, a group of order 39 cannot have a normal Sylow 13-subgroup. This is because the number of Sylow 13-subgroups must divide 39 and leave a remainder of 1 according to Sylow's first theorem. Since 13 does not divide 39, there cannot be a normal Sylow 13-subgroup.

## How do Sylow theorems help in determining the structure of a group?

Sylow theorems provide information about the number of subgroups in a group, which in turn can help in determining the structure of the group. For example, if the number of Sylow p-subgroups is 1, then the group is a p-group and has a cyclic subgroup of order p. Sylow theorems also help in finding normal subgroups and the intersection of subgroups, which are important in understanding the structure of a group.

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