The wikipedia page for group with operators makes the following claim about composition series as being analogous to compactness:
The Jordan–Hölder theorem also holds in the context of operator groups. The requirement that a group have a composition series is analogous to that of compactness in topology, and can sometimes be too strong a requirement. It is natural to talk about "compactness relative to a set", i.e. talk about composition series where each (normal) subgroup is an operator-subgroup relative to the operator set X, of the group in question.
Does anybody have any comments on this, preferably with links/references? Gonna require outside-the-box thinking on this one, but the intro to Zassenhaus' group theory motivating taking composition series is definitely in the spirit of compactness:
The idea of Holder for solving this problem was later made a general principle of investigation in algebra by E. Noether. We are referring to the consistent application of the concept of homomorphic mapping. With such mappings one views the objects, so to speak, through the wrong end of a telescope. These mappings, applied to finite groups, give rise to the concepts of normal subgroup and of factor group. Repeated application of the process of diminution yields the composition series, whose factor groups are the finite simple groups. These are, accordingly, the bricks of which every finite group is built.
when you view compactness as a local-to-global principle.
Thanks!