Inducing characters and nilpotent injectors. (English) Zbl 1003.20006
Let \(H\) be a subgroup of a finite group \(G\), let \(\text{cf}(H)\) denote the space of complex class functions on \(H\), and let \(\text{cf}(H)^G\) denote the space of class functions on \(G\) induced from \(H\). A basis of the space \(\text{cf}(H)^G\) is called “good” if the basis vectors are induced from irreducible characters of \(H\) and every character of \(G\) induced from \(H\) is a nonnegative integer combination of the basis vectors. A good basis does not always exists, but when it does it is unique, and determines a canonical partition of \(\text{Irr}(G)\) analogous to Brauer’s \(p\)-blocks (the latter being obtained by taking for \(H\) a \(p\)-complement of the \(p\)-soluble group \(G\)).
G. Navarro proved [in J. Aust. Math. Soc., Ser. A 61, No. 3, 369-376 (1996; Zbl 0874.20006)] that if \(G\) is soluble and \(H\) is a nilpotent injector of \(G\), then a good basis for \(\text{cf}(H)^G\) exists. The present paper refines that result as follows: if \(H/N\) is a nilpotent injector of some soluble quotient \(G/N\) of \(G\), then a good basis for \(\text{cf}(H)^G\) exists.
G. Navarro proved [in J. Aust. Math. Soc., Ser. A 61, No. 3, 369-376 (1996; Zbl 0874.20006)] that if \(G\) is soluble and \(H\) is a nilpotent injector of \(G\), then a good basis for \(\text{cf}(H)^G\) exists. The present paper refines that result as follows: if \(H/N\) is a nilpotent injector of some soluble quotient \(G/N\) of \(G\), then a good basis for \(\text{cf}(H)^G\) exists.
Reviewer: Sandro Mattarei (Povo)
MSC:
20C15 | Ordinary representations and characters |
20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |
Keywords:
finite groups; complex class functions; good bases; induced characters; irreducible characters; soluble groups; nilpotent injectorsCitations:
Zbl 0874.20006References:
[1] | Navarro, J. Austral. Math. Soc. Ser. A 66 pp 104– (1999) |
[2] | Navarro, J. Austral. Math. Soc. Ser. A 61 pp 369– (1996) |
[3] | DOI: 10.1007/BF02785588 · Zbl 0958.20015 · doi:10.1007/BF02785588 |
[4] | Isaacs, Character theory of finite groups (1994) · Zbl 0849.20004 |
[5] | DOI: 10.1007/BF02771470 · Zbl 0228.20005 · doi:10.1007/BF02771470 |
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