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Characters vanishing on all but two conjugacy classes. (English) Zbl 0536.20005

A theorem of Burnside asserts that for any group \(G\) any non-linear irreducible character has zero (deep generalizations of this result were obtained by P. X. Gallagher, and also (in unpublished papers) by E. M. Zmud’ and A. J. Weitzblit). The extreme case is considered here, namely, groups \(G\) for which a character \(\chi\) exists which vanishes on all but two conjugacy classes. If \(| G|>2\) than \(\chi\) is unique and is, moreover, the unique faithful irreducible character of \(G\). In this case \(G\) contains a unique minimal normal subgroup \(N\) which is an elementary abelian \(p\)-group, \(p\) is a prime. The character \(\chi\) vanishes on \(G-N\) and \(\chi(n)\neq 0\) for all \(n\in N\) (i.e. \(N^{\#}\) is a conjugacy class of \(G\)). We have \(C_ G(N)=O_ p(G)\). If \(G\) is solvable then \(O_ p(G)=F(G)\), the Fitting subgroup of \(G\). Moreover, \(G/O_ p(G)\) has a normal \(p\)-complement which is isomorphic to the multiplicative group of a near-field, and a Sylow \(p\)-subgroup of \(G/O_ p(G)\) is abelian.
Theorem 5.6. Let \(G\) be a non-solvable group with the title property. Then \(O_ p(G)\neq \{1\}\) for some unique prime \(p\). Moreover, the group \(H=G/O_ p(G)\) has one of the following forms:
(i) There exists \(S\triangleleft H\) with \(S\cong SL(2,q)\), where \(q>2\) is a power of \(p\), \(H/S\) is a cyclic \(p\)-group and \(C_ H(S)=Z(S)\).
(ii) \(p=3\), \(H\) contains a normal subgroup S of index 2, and \(S\cong SL(2,5)\) and \(C_ H(S)=Z(S)\). \(S\) is not split in \(H\) (i.e. \(S\) contains all involutions from \(H\)).
(iii) \(p=3\) and \(H\cong SL(2,13)\).
(iv) \(p=11\) and \(H\cong SL(2,5)\).
(v) \(p=29\) and \(H\cong SL(2,5)\times C_ 7\).
(vi) \(p=59\) and \(H\cong SL(2,5)\times C_{29}\).
Theorem 6.2. \(N=O_ p(G)\) iff \(G\) is a doubly transitive Frobenius group (we assume \(| G|>2\)).
Theorem 6.3. Let \(Q\) be any \(p\)-group and let \(a>1\) be any integer. Then a group \(G\) exists satisfying the following conditions: (a) \(G\) has an irreducible character which vanishes on all but two conjugacy classes. (b) \(G=PH\) where \(P\triangleleft G\) is a Sylow \(p\)-subgroup and \(H\) is cyclic of order \(p^ a-1\). (c) \(Z(P)H\) is a doubly transitive Frobenius group of order \(p^ a(p^ a-1)\). (d) \(Q\) is isomorphic to a subgroup of \(P/Z(P)\).
We mention that E. M. Zmud’ in a series of papers studied finite groups \(G\) with the following property: \(G\) has an irreducible non-linear character \(\chi\) such that all zeros of \(\chi\) are conjugate in \(G\).
Reviewer: Ya.G.Berkovich

MSC:

20C15 Ordinary representations and characters
20D05 Finite simple groups and their classification
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
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