# Thread: classify groups order 77

1. ## classify groups order 77

This is a neat question [there is a fast solution]:

Classify, up to isomorphism, groups of order 77.

2. Originally Posted by GaloisTheory1
This is a neat question [there is a fast solution]:

Classify, up to isomorphism, groups of order 77.
My attempted solution is as follows:

1. $Z_{77} = Z_{7} \times Z_{11}$.

Since 11 is not congruent to 1modulo7, it has one Sylow 7-subgroup and one Sylow 11-subgroup, which has an intersection at {e}. Thus, the group of order 77 is the direct product of a normal Sylow 7-subgroup and a normal Sylow 11-subgroup, which is cyclic.

I think there is only one group of order 77 up to isomorphism.

3. Originally Posted by GaloisTheory1
This is a neat question [there is a fast solution]:

Classify, up to isomorphism, groups of order 77.
As aliceinwonderland said this is a special case of $|G| = pq$ where $p,q$ are odd primes.
There some conditions on $pq$ (what alice did) that force $G\simeq \mathbb{Z}_{pq}$.