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A group G is called completely reducible if it is a direct product of simple groups. It is known that a Krull-Schmidt Remak type unicity for the decomposition in the direct product of simple groups hold. The proof s of this facts can be found for example in Chapter 3 of D. Robinson's book A Course in the Theory of Groups

My questions are the following:

  1. Are any known characterizations for finite completely reducible groups given in the literature? Given a finite group how can one decide whether the group is completely reducible or not?, of course without searching for direct product decompositions if possible.

  2. If one replaces the direct product decomposition with iterated crossed product decomposition is anything known about unicity in this case?

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  • $\begingroup$ Nothing else known about completely reducible groups? $\endgroup$
    – anonymus
    Commented Mar 6, 2011 at 9:33

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Well, it depends what you are prepared to accept as a reasonable answer. A finite group is completely reducible if and only if the intersection of all its maximal normal subgroups is trivial. This is presumably a lot of work to check for any reasonably sized group. On the other hand, you can regard the intersection of the maximal normal subgroups as a kind of "radical" of a finite group, and the above characterization seems reasonable. One (fairly obvious) fact to use in proving that a finite group is NOT completely reducible is that if G is completely reducible, then so are all its non-trivial normal subgroups. Checking minimal normal subgroups is no help here, of course.

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