Why do we care about simple groups rather than indecomposable groups?

Question

A first thing that we do to analyze something is "divide and conqur." So it is natural to consider simple groups. In the same way, it appears to be natural to consider indecomposable groups. However, no one taught me about indecomposable groups and I know it quite recently.

Of course, indecomposable groups is larger class than simple groups. Having said that, it doesn't answer my question fully because a lager concept is sometimes easier to deal with (for example, $\mathbb{Z} \subseteq \mathbb{R}$).

Is there any reason that we care about simple groups rather than indecomposable groups?

Added: From the comment to DustanLevenstein — For example, in module theory, indecomposable modules looks be treated equal with simple modules. importance of indecomposable modules and simple modules seems be at the same level. What's the difference between these?

Answer 1

Making good mathematical definitions involves contemplating a tradeoff between generality and power. For example, monoids are more general than groups, but one reason people usually don't learn about monoids in a first course on abstract algebra is that it's hard to prove anything substantive about even, say, finite monoids without further assumptions, whereas groups are less general but one can prove lots of interesting things about, say, finite groups (e.g. Lagrange's theorem, the Sylow theorems).

Indecomposable groups are certainly more general than simple groups, and it is true that if we understood indecomposable groups then understanding all groups would be straightforward, but the difficulty hasn't been magically removed. The point of Dustan Levenstein's comment is that because it's so easy to classify all finite groups once we've classified all finite indecomposable groups, the latter should be essentially as hard to understand as the former. In particular one should not be able to make much stronger claims about indecomposable groups than one would be able to make about arbitrary groups: it's a bad tradeoff of generality vs. power.

Module theory is closer to linear algebra than to (nonabelian) group theory and so one can hope for a closer relationship between indecomposability and simplicity in this case. I don't have more specific knowledge about what happens in this case though.