 Solution

ID: past-exam-of-the-introduction-to-representation-theory-course-of-the-university-of-oxford/2023/1/a/i/solution

A

-module is completely reducible if it is a direct sum of modules each of which is a simple module:

V = ⨁_{j \in J} S_{j} .

Here a simple module is nonzero and has no submodules other than zero and itself. This is precisely the definition of a semisimple module. The indexing set need not be finite unless

V

has finite dimension; the zero module is included as the empty direct sum of modules.

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