can someone give me an example of this theorem:
a group G, where G is not {1}, with no non trivial subgroup is a finte cyclic group of prime order.
i was reading the proof for this theorem..and i understand how this theorem is proven but im not really sure why this is even possible. an example that satisfy this theorem would be really healpful.
If you just want an example, consider the integers under addition modulo p, for some prime p-- p=3 would do.
I sympathize with your need for examples. When I was taught algebra (long ago) there were very few examples presented, and I never felt I knew what was going on. I hope the style of teaching has changed.