On $p$-adic congruence of some class functions on a finite group

Content Provider	Semantic Scholar
Author	Blau, Harvey I.
Copyright Year	1984
Abstract	Certain p-adic integer-valued class functions on a finite group are shown to be congruent modulo a suitable power of p. This is applied to prove and extend a result of Plesken on central characters of a p-block with cyclic defect group. G is an arbitrary finite group, p a fixed rational prime, K a p-adic number field of characteristic zero, R the ring of integers in K, 7r a generator of J(R), and R = R/7rR. Assume that K and R are splitting fields for all subgroups of G. Let v be the p-adic valuation on K, scaled so that Iv(p) = 1. If X is an R-valued class function on G with x(l) #& 0 (as is the case if X is an ordinary character of G), g is an element of G, and gG denotes the conjugacy class of g, we define wXy(g) = X(9) 1G1/X( ) Then wx is a K-valued class function on G. If X is an irreducible character, then W. is R-valued, and is a central character of KG. Irreducible characters Xi and X2 are in the same p-block if and only if wxI (9)-WX2 (9) (mod 7rR) for all g E G [1, IV.4.2]. Plesken [2, VIII.4] has shown (as an application of results on R-orders in the group ring RG) that if B is a p-block of G with cyclic defect group of order pa, and if Xu, Xv are nonexceptional irreducible characters in B which occupy nodes in the Brauer tree for B which are not separated from each other by the exceptional node, then (1) wu(g) cwx,(g) (modpaR), for all g E G. Our purpose here is to show, by elementary methods of modular representation theory and well-known facts about blocks with cyclic defect groups, that (1) holds for all nonexceptional characters in B. This will be a consequence of the following general result. THEOREM. Let ED be an R-linear combination of characters of projective RGmodules. Suppose that 1? = s + X, where . X are R-valued class functions on G such that ~(')X(') 78 0. Let V(IGI) = n and v(x(1)) = m. Assume that m V(ICG(g)I) + v(lgGI) = n [1, IV.2.5]. So in either case, X(j)4_(g) I gGj I0 (mod pn+mR). Received by the editors January 3, 1984. 1980 Mathematics Subject Classification. Primary 20C,15, 20C20; Secondary 20C11. 0)1984 Amiierican Mathemiiatical Society 0002-9939/84 $1 00 + $.25 per page 485 This content downloaded from 40.77.167.54 on Thu, 06 Oct 2016 04:32:38 UTC All use subject to http://about.jstor.org/terms
Starting Page	485
Ending Page	486
Page Count	2
File Format	PDF HTM / HTML
DOI	10.1090/S0002-9939-1984-0760930-9
Volume Number	92
Alternate Webpage(s)	https://www.ams.org/journals/proc/1984-092-04/S0002-9939-1984-0760930-9/S0002-9939-1984-0760930-9.pdf
Alternate Webpage(s)	https://doi.org/10.1090/S0002-9939-1984-0760930-9
Language	English
Access Restriction	Open
Content Type	Text
Resource Type	Article