ABSTRACT
Most groups besides the trivial group {e} possess a rich “internal structure.” Indeed, all non-trivial groups, whether finite or infinite, whether Abelian or non-Abelian, have smaller groups inside of them. It is very important to remember that a subgroup must possess the exact same operation as that of the group containing it. Finite groups can possess non-trivial proper subgroups just as infinite groups do, but these situations may require a closer look. A very rich source of subgroups of a group G is the collection of cyclic groups contained within G.
