= Relationship between the quotient group and direct products
Although quotients look a bit real number division, there are some important differences with the "group analog of multiplication" of <direct product of groups>.
If a group is isomorphic to the <direct product of groups>, we can take a quotient of the product to retrieve one of the groups, which is somewhat analogous to division: https://math.stackexchange.com/questions/723707/how-is-the-quotient-group-related-to-the-direct-product-group
The "converse" is not always true however: a group does not need to be isomorphic to the product of one of its <normal subgroups> and the associated <quotient group>. The wiki page provides an example:
\Q[Given G and a normal subgroup N, then G is a group extension of G/N by N. One could ask whether this extension is trivial or split; in other words, one could ask whether G is a direct product or semidirect product of N and G/N. This is a special case of the extension problem. An example where the extension is not split is as follows: Let $G = Z4 = {0, 1, 2, 3}$, and $ = {0, 2}$ which is isomorphic to Z2. Then G/N is also isomorphic to Z2. But Z2 has only the trivial automorphism, so the only semi-direct product of N and G/N is the direct product. Since Z4 is different from Z2 × Z2, we conclude that G is not a semi-direct product of N and G/N.]
TODO find a less minimal but possibly more important example.
This is also semi mentioned at: https://math.stackexchange.com/questions/1596500/when-is-a-group-isomorphic-to-the-product-of-normal-subgroup-and-quotient-group
I think this might be equivalent to why the <group extension problem> is hard. If this relation were true, then taking the direct product would be the only way to make larger groups from normal subgroups/quotients. But it's not.