Zeros of Characters of Finite Groups
نویسندگان
چکیده
There are some variations of theorems A and B which are simply not true. For instance, if χ in Irr(G) has degree divisible by p, then there does not necessarily exist a p-element on which χ vanishes. It is enough to consider L2(11) with any character of degree 10 and p = 2. It is also not true that if χ vanishes on some element x, then χ has to vanish on some p-part of x. For instance, if G is M11, then χ has an irreducible character of degree 11, vanishing on an element of order 6 and which is nonzero on 2and 3-elements. Thirdly, it is not true that a nonlinear character has to vanish on some element of prime order, as shown by any quaternion group. Interestingly enough, this seems to be the case for simple groups (and we do prove this for the groups of Lie type and the sporadic groups).
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