I've been studying algebra and I came upon the assertion that simple groups are important because they're "building blocks", similar to prime numbers. But if we take the cyclic group of order four as an example, it has the two element group as both a normal subgroup and a quotient but taking that product gets us a different group back, not the original cyclic group.
So I guess my question is how does finding normal subgroups help us understand/simplify a group? Or is there more significance to simple groups?
TIA