On a Theorem of J. A. Green

Transcription

1 JOUNL OF LEB 209, TICLE NO J On a Theorem of J reen Kench Yamauch Department of Mathematcs, Facult of Educaton, Chba Unerst, Yaocho, Chba , Japan E-mal: amauch@mathechba-uacjp Communcated b Walter Fet eceved prl 1, 1998 DEDICTED TO POFESSO KTSUMI SHITNI J reen proved a theorem whch s the converse to a theorem of Brauer ŽProc Camb Phlos Soc 51 Ž 1955, In ths artcle we gve another proof of the theorem concretel wthout usng Frobenus s formula for nduced characters and we also state some comments on Brauer s nducton theorem 1998 cademc Press 1 INTODUCTION Throughout ths artcle,, Z, and C denote a fnte group, the rng of ratonal ntegers and the feld of complex numbers, respectvel For a fnte set S we denote the number of elements n S b S and for a ratonal prme p we denote the p-part of Sb S p Let 1 Ž the prncpal character,, 4 1 h be the full set of nonsomorphc rreducble complex characters of Let charž be the rng of h generalzed characters of That s, char Ý a a Z Ž 1 1,,h 4 Then charž s a subrng of the rng cfž of all complex-valued class functons on From now on, for an subrng of C, we denote b char Ž the set of -lnear combnatons of the complex characters of for smplct For cfž and a subgroup H of, H denotes the restrcton of to H Let H be a faml of subgroups of Then we consder the followng three statements wth respect to H Ž If for an cfž, H charž H for all H H, then charž $2500 Coprght 1998 b cademc Press ll rghts of reproducton n an form reserved 708

2 ON THEOEM OF J EEN 709 Ý 4 4 H H char H char, where char H s the set of all generalzed characters of of the form Žthe generalzed character of nduced b wth charž H Each elementar subgroup of s contaned n some conjugate of some subgroup belongng to H However, as s well known these statements are equvalent proof of Ž s obtaned b Brauer s proof of Theorem 3 n 1 and b usng two formulas Ž of Ž 385 Theorem n 2 and 1 ÝaHH where ah Z, charž H and H H In 3 J reen gves a proof of H b usng Frobenus s formula for nduced characters, n order to prove Ž Ž that s, the converse to a theorem of Brauer In ths artcle we ntend to gve a proof of Ž whch s dfferent from reen s proof, b makng an applcaton of the characterstc class functons of n applcaton of the characterstc class functons of a fnte group s one of the basc tools n the stud of the structure of a character rng of a fnte group and such applcatons are also stated n 5 and 6 Let be an subrng of C consstng of algebrac ntegers such that Z Then we state the followng statement whch s smlar to Ž, wth respect to a faml H of subgroups of Ž v If for an cfž, H char Ž H for all H H, then char Ž We note that a proof of Ž v s the same as that of Ž, because we can see easl that Brauer s proof of Theorem 3 n 1 gves a proof of Ž v Ž Hence we have Ž v In what follows, we assume further that where s a prmtve th root of unt Then for provng Ž Ž, t s suffcent to prove Ž v Ž 2 POOF OF v Let C,,C 4 be the full set of conjugate classes of and c 1 Ž 1 h 1 the dentt element of,,ch be the representatves of C 1,,C h, respec- tvel Then we defne class functons f Ž 1,,h of as follows fž c 1, fž cj 0, Ž j We sa that these class functons are the characterstc class functons of and that f corresponds to C or C corresponds to f Ž 1,,h B

3 710 KENICHI YMUCHI the second orthogonalt relaton for the group characters, we have 1 f Ž c, Ž 1,,h h Ý CŽ c j1 j j Because Ž c 1 1 for all 1,,h, we have LEMM 21 Keepng the precedng notatons we hae Ž C Ž c f char Ž Ž 1,,h mf char Ž for m Z, 1 m C Ž c Ž 1,,h Let p be a ratonal prme and consder the p-elementar group ² x: P where x s a p -element of a fnte group and P s a Slow p-subgroup of C Ž x Let H be a subgroup of whch does not contan an conjugate of ² x: P, and let be the characterstc class functon of whch corresponds to the conjugate class C of contanng x Let us set C Ž x mp k, Ž m, p 1, k 1 Then we have LEMM 22 In the preous stuaton, mp H belongs to char Ž H Proof If C H Ž an empt set, then t s clear that H 0 Hence we have mp H char Ž H Therefore we ma assume that C H Let D 1,, Dt be the dstnct H-conjugate classes contaned n C H, and let h D, Ž 1,,t Let be the characterstc class functon of H whch corresponds to D, Ž 1,,t Then each h s -conjugate to x and so there exsts such that h x Further we have ² h : P Ž ² x: P C Ž x C Ž h, Ž 1,,t Here we show that C h p, Ž 1,,t H p ssume b wa of k contradcton that C h p Then we have C Ž h C Ž h H p H p p k C x p Let P be a Slow p-subgroup of C Ž h p H Then P s C h -conjugate to P Therefore there exsts z C Ž h such that Ž z z P P P Hence we have z z z H ² h : P ² h : P ² h : P C Ž h, and so ² x: P s -conjugate to ² h : P Ths means that H contans a conjugate of ² x: P Ths s contrar to our assumpton Therefore we have C h p, Ž 1,,t H p as clamed k Because C h p p C Ž x and C Ž h C Ž h H p p H H, we can see that C x p mp s dvsble b C Ž h H Here we set n mp C h, 1,,t Then we have mp H H

4 ON THEOEM OF J EEN 711 t Ý n C Ž h Because C Ž h char Ž H, Ž 1,,t 1 H H b Lemma 21, we have mp H char Ž H as clamed Thus the proof s complete Proof of Ž v Ž Let H be a faml of subgroups of a fnte group whch satsfes Ž v and let ² x: Pbe a p-elementar subgroup of for a k p -element x and a p-group P Let us set C x mp, Ž m, p 1, k 1 Let be the characterstc class functon of whch corresponds to the conjugate class of contanng x ssume b wa of contradcton that ² x: P s contaned n no conjugate of a subgroup belongng to H Then the precedng lemma mples that mp H belongs to char Ž H for all H H Because H satsfes v, t follows that mp char Ž Ths s contrar to Lemma 21Ž Therefore ² x: P s contaned n some conjugate of some subgroup belongng to H and the result follows Fnall we restate a theorem concernng the statements Ž Ž v rng theoretcall Let be a rng wth Z C and let H be a faml of subgroups of Then we defne Ž, H and I Ž, H as follows, H the set of class functons cf char Ž H for all H H, such that H 4 I, H Ý char H H H If Z, we delete the subscrpts and we wrte Ž, H and IŽ, H B Ž 82 Lemma n 4, t follows that Ž, H s a rng n whch I Ž, H s an deal and from the defntons we have I Ž, H IŽ, H charž Ž, H Furthermore, f 1 IŽ, H for some faml H, t follows that I Ž, H char Ž Ž, H for all wth Z C Because the statements Ž Ž v are equvalent, we have THEOEM 23 Let H be a faml of subgroups of and let be an subrng of C consstng of algebrac ntegers such that Z Then the followng condtons are equalent: Ž char Ž Ž, H I IŽ, H each elementar subgroup of s contaned n some conjugate of some subgroup belongng to H emark We cannot extend the rng n Theorem 23 anmore We gve a counterexample

5 712 KENICHI YMUCHI 1 Let B be an subrng of C such that Z B and 2 B and let S4 be the smmetrc group on four letters We set 1 Ž 1 the dentt element of S, Ž 12, Ž 12Ž 34, Ž 1 2 3, and Ž 1234We also set H ² : Ž the cclc subgroup generated b Ž 1, 2, 3, 4, 5 Let be the prncpal character of H and be the character of S4 whch s nduced b Ž 1, 2, 3, 4, 5 Then we have S Hence 1 I Ž S, H where H H, H, H, H, H 4 S B Ths mples 4 that char Ž S Ž S, H But 1 IŽ S, H B 4 B 4 S 4 because a Slow 2-sub- 4 group of S4 s contaned n no conjugate of a subgroup belongng to H EFEENCES 1 Brauer, characterzaton of the characters of groups of fnte order, nn of Math 57 Ž 1953, C W Curts and I ener, epresentaton Theor of Fnte roups and ssocatve lgebras, Wle-Interscence, New York, J reen, On the converse to a theorem of Brauer, Proc Camb Phlos Soc 51 Ž 1955, I M Isaacs, Character Theor of Fnte roups, cademc Press, New York, K Yamauch, On somorphsms of a Brauer character rng onto another, Tsukuba J Math 20 Ž 1996, K Yamauch, On automorphsms of a character rng, Tsukuba J Math 20 Ž 1996,