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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!

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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. –  Mark Bennet Nov 14 '12 at 15:58
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up vote 6 down vote accepted

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)

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that really helps, thanks a ton! –  Chloe.H Nov 19 '12 at 2:17
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