Unfaithful minimal Heilbronn characters of \(L_2(q)\). (English) Zbl 1263.20004
Summary: When a minimal Heilbronn character \(\theta\) is unfaithful on a Sylow \(p\)-subgroup \(P\) of a finite group \(G\), we know that \(G\) is quasi-simple, \(p\) is odd, \(P\) is cyclic, \(N_G(P)\) is maximal and either \(N_G(P)\) is the unique maximal subgroup containing \(\Omega_1(P)\) or \(G/Z(G)\cong L_2(q)\) for \(q\) an odd prime with \(p\) dividing \(q-1\). In this paper we examine the exceptional case, where \(G/Z(G)\cong L_2(q)\), explicitly constructing unfaithful minimal Heilbronn characters from the non-principal irreducible characters of \(G\).
MSC:
| 20C15 | Ordinary representations and characters |
| 20D06 | Simple groups: alternating groups and groups of Lie type |
| 20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |
| 11M06 | \(\zeta (s)\) and \(L(s, \chi)\) |
| 11M26 | Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses |
Keywords:
Heilbronn characters; Artin conjecture; monomial characters; finite simple groups; virtual characters; irreducible charactersReferences:
| [1] | Group representation theory, Part A 7 (1971) |
| [2] | DOI: 10.1080/00927879708825877 · Zbl 0874.11074 · doi:10.1080/00927879708825877 |
| [3] | DOI: 10.1016/j.jalgebra.2010.09.043 · Zbl 1243.20010 · doi:10.1016/j.jalgebra.2010.09.043 |
| [4] | K (1998) |
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