8
$\begingroup$

Ok,

so this was a recent question I've been asked for homework. "Classify all groups of order $1225$". That's all the question says. How in the world do I approach this?? We have the symmetric group of order $1225$, cyclic group of order $\,1225 = 5*5*7*7\,$ and so we have that as a group.. but the list just seems so big. How do we classify all groups of this order, it's huge! We had a similar question on groups of order $8$ the other week and it wasn't easy..

Thanks for any help guys!

  • 0
    Such questions can very often be addressed using knowledge of the classification of Abelian Groups, and Sylow's theorems to get a handle on possible nonabelian groups.2012-11-14

1 Answers 1

7

Hints:

1) Prove there are exactly one Sylow 5-subgroup and one Sylow 7-subgroup

2) Prove that every group of order the square of a prime number is abelian

3) Show thus that any group of order $\,1225\,$ is abelian and thus

4) There are only $\,4\,$ groups of order $\,1225\,$ (Using the Fundamental Theorem of Finitely Generated Abelian Groups)

  • 0
    that really helps, thanks a ton!2012-11-19