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.
