It will become much easier to tell if a group is simple or not once you learn Sylow-theory. Sylow-theory ("see-low theory") is really just an awesomely powerful theorem, and it's primary use is to prove whether or not all groups of a certain order are simple (at least in my experience).Other than that, I know of no way to simply tell if a group is simple or not. Obvioiusly any abelian group with a proper non-trivial subgroup is not simple.

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