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#### buckeye1973

##### New member

- Feb 22, 2012

- 5

I'm looking for basic strategies for identifying the subgroups of a group. I believe I have to use conjugacy classes and cycle types, but I'm not sure how to apply those concepts.

Let me pose a specific problem:

Let $G$ be a subgroup of the symmetric group $S_5$, with $|G| = 4$.

By looking this up on a chart, I see that there are three subgroups of $S_5$ of order 4, two Klein-4 groups and the cyclic group $C_4$, so $G$ must be isomorphic to one of these.

How would I go about showing that $G$ must be isomorphic to one of these without looking it up? I'm hoping for a general algorithm here, if possible, such that I could also find subgroups of a certain order of given dihedral groups, for example.

Thanks!

Brian