An object $X$ in a category $\cal C$ with an initial object is called indecomposable if $X$ is not the initial object and $X$ is not isomorphic to a coproduct of two noninitial objects.

A group $G$ is called indecomposable if it cannot be expressed as the internal direct product of two proper normal subgroups of $G$. This is equivalent to saying that $G$ is not isomorphic to the direct product of two nontrivial groups.

These definitions give rise to an unfortunate-sounding result: there are objects in the category Grp which are indecomposable (as objects) but are decomposable (as groups). An example is $\Bbb Z_2\times\Bbb Z_2$.

My question is what events and what needs culminated in this unfortunate state of affairs? That is, what is the background of the term 'indecomposable' in group theory and category theory? And why was this term settled on in both subjects?


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