Figuring out what subgroups a group contains is one way to understand its structure. For example, the subgroups of Z6 are {0}, {0, 2, 4} and {0, 3}—the trivial subgroup, the multiples of 2, and the multiples of 3. In the group D6, rotations form a subgroup, but reflections don’t. That’s because two reflections performed in sequence produce a rotation, not a reflection, just as adding two odd numbers results in an even one.
Certain types of subgroups called “normal” subgroups are especially helpful to mathematicians. In a commutative group, all subgroups are normal, but this isn’t always true more generally. These subgroups retain some of the most useful properties of commutativity, without forcing the entire group to be commutative. If a list of normal subgroups can be identified, groups can be broken up into components much the way integers can be broken up into products…
