chapter  19
14 Pages

Induced Characters

WithVictor E. Hill, Thomas T. Read

In Chapter 13, we used representations of a factor group G/N to find representations of the group G. It is also possible, although a bit more complicated, to lift a representation of a subgroup H of G to a representation of G itself. If you reflect, you’ll realize that this second construction is naturally more difficult: instead of “burying” a bit of the given group G in a kernel N, we look at a subgroup H which is “surrounded” by Terra Incognita in the group G. Yet it turns out that this construction is a useful and powerful tool for finding new representations, in particular those that are irreducible. As a result, we’ll be able to complete the project, begun at the end of Chapter 15, of constructing the character table of the alternating group A 5 of order 60. All groups in Chapter 19 will be assumed to be finite.