Vol. 55, No. 1, 1974

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Characters fully ramified over a normal subgroup

Stephen Michael Gagola, Jr.

Vol. 55 (1974), No. 1, 107–126

Let H be a group and N a normal subgroup. Assume that χ is an irreducible (complex) character of H, and that the restriction of χ to N is a multiple of some irreducible character of N, say 𝜃. Then χN = e𝜃, and e is called the ramification index. It is easy to see that it always satisfies e2 |H : N|, and when equality holds, χ is said to be fully ramified over N. It is this “fully ramified case” which will be studied here in some detail. As an application of some of the methods of this paper, we prove the following solvability theorem in the last section. If H has an irreducible character fully ramified over a normal subgroup N and if p4 is the highest power of p dividing |H : N| for all primes correspondin g to nonabelian Sylow p-subgroups of H∕N, then H∕N is solvable.

Mathematical Subject Classification 2000
Primary: 20C15
Received: 30 April 1973
Revised: 30 July 1974
Published: 1 November 1974
Stephen Michael Gagola, Jr.