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Representations and Characters of Groups, Second Edition by Martin Liebeck, Gordon James

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17

Normal subgroups and lifted characters

If N is a normal subgroup of the finite group G, and N ≠ {1}, then the factor group G/N is smaller than G. The characters of G/N should therefore be easier to find than the characters of G. In fact, we can use the characters of G/N to get some of the characters of G, by a process which is known as lifting. Thus, normal subgroups help us to find characters of G. In the opposite direction, it is also true that the character table of G enables us to find the normal subgroups of G; in particular, it is easy to tell from the character table whether or not G is simple.

The linear characters of G (i.e. the characters of degree 1) are obtained by lifting the irreducible characters of G/N in the case where N

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