ABOUT A CLASS OF FINITE GROUPS

İstanbul Üniv. Fen Fak. Mat. Der., 54 (1995), 25-27 25 ABOUT A CLASS O F FINITE GROUPS Ion A R M E A N U University of Bucharest, Physics Faculty, Mathematics Dept., Bucharest-Magurele, P.O. Box MG-11, ROMANEA Summary : This note is concerned to prove some interesting facts about the groups G who have the property that NG (o*) is subnormal in JVc (a) for every a^x e G such that ax = xa, where a has odd order and the order of x is a power of 2. BİR S O N L U G R U P S I N I F I H A K K I N D A Özet : Bu çalışmada şu özeliği taşıyan sonlu G gruplarına İlişkin bazı ilginç sonuçlar elde edilmektedir: "a,xeG, a nın mertebesi tek, jcin mertebesi 2 nin kuvveti biçiminde ve ax — xa olmak üzere, her a, x çifti için NG (ax), NG (a) run bir normal alt grubudur". I n this note we will use only finite groups and the notations and definitions will be those o f [3]. Definitions : a) We will say that a group G is an ,4-group i f for every a e G o f odd order and for every x e G o f order a power o f 2, such that ax = xa, then N (ax) is subnormal i n N (a). G G b) We will say that a group is a Q-group i f all its irreducible characters are rational valued. c) We will say that a group is a Q - g r o u p i f it is a Q-group and an ^4-group too. Proposition 1. Then: Let G be an ,4-group. Let a, xe G be as i n the Definitions. a) N (ax) ^ N (a). b) C (ax) is subnormal in N (a). G G G G c) Let H be a 2-Sylow group of N (a) and H be the 2-Sylow group of C (a) such that ff ^H. Then Hn N (ax)=N is a 2-Sylow group of N (ax) and ff n C (ax) = C is a 2-Sylow group o f C (ax). G G 0 Q a G 2 0 z G G 26 Ion ARMBANU The proof is obvious. For the next we will use the already introduced notations. Theorem 2. Proof. Let G be a Q^-group. Then H < H is fusion free. 0 Let be xeH^ and b&H\'H- . We will show>• that there exists a Q heH such that b'^xb = h~^xh. Let u e A u t (d) be the nontrivial inner automorphism given by b, where A u t ( a ) is the 2-Sylow group o f A u t (a). Since G has rational valued characters we have that 0 2 2 1 N (ax)lC (ax)=*Aut(ax)=*Aut(a)xAut(x) G (see [3]ypg 11) and that G N IC ~N . 2 2 C(ax)/C(ax) 2 ; (see [4], pg. 56). Therefore there exists a b e N u 2 such that its image in A u t (a) X A u t (x) to 2 2 l be «.1, consequently: b~ ab - = b~} ab and: b cominutes iwith x. u u Since b and b lead to the same automorphism of <a>, there exists a such that b =b h. Then . ,. u hEH 0 u l B~ xb Theorem 3. l l = h- b-' xb l h = h~ xh . " u Let G be a Q^-group. Then ff Q is a Q-group. 1 Proof. Let f : N (a) - > A u t (a) be given by f (x) (a) = x ax* . Since G has rational valued characters, f is an epimorphism. Let heH , and let z, w be the generators o f A u t (h). For a G a a fah '• N (ah) 0 A u t (ah) ^ A u t (a) x A u t (h) G there exist x,yeN (ah) such that f (x) = z and f (y) ~ w. Since A u t (A) is a 2-group i t follows that any odd powers of z and w are generators for A u t (A) too. Therefore i f | x\=2i q and |_y|=2 r, with q and /* odd integers, considering x =x and j = y it follows h a t . / ^ ( x ^ and / ^ ( j j are generators for A u t (A), besides h, x ,y sC(a) n N(ah). Since G is an ^4-group, using the Sylow's theorem we obtain that there exist u, veC(a) n N(ah) such that the elements x ^ux u~ and y =vy v~ belong-to the 2-Sylow group H r\ N(ah) of N(ah). Besides f (x ) = f (x ) and f (y ) = f (ydTherefore f (x ) and f (y ) generate A u t (A). G ah ah ft q T l x l 1 1 2 1 1 2 ah 2 1 ah 1 Q a!l 2 ah ah 2 a/l 2 Remark. I n particular, for a = 1 we obtain that for a Q^-group it holds the old standing conjecture (see [3], pg. 13) that asserts that for a Q-group the 2-Sylow subgroups are Q-groups too. I n fact, at this moment I do not know an example of a Q-group which is not an ^4-group. ABOUT A CLASS O F F I N I T E GROUPS Proposition 4. a) Let G be an A-gvoup and H^G. 27 Then H is also an ,4-group. b) Let G be a Q^4-group and / f ^ G fusion free. Then J î i s also a Q^4-group. Proof. We know that N (ax) is subnormal in N (a). Then N (ax) = = N (ax) n H is subnormal in N (a) = N (a) n H (see [4], pg. 127). G G G H Proposition 5. H G Let G be a Qv4-group with abelian 2-Sylow group. Then: a) A n y 2-Sylow group is isomorphic with Z x . . . x Z m (X) = I , y ^ e l r r (G). 2 2 and the Schur index & b) G is strong real. Proof. Since the 2-Sylow groups are abelian Q-groups i t follows immediately that they are isomorphic with Z X . . . x Z . Through the Brauer-Speiser Theorem (see [5], pg. 9) and Fein-Yamada Theorem (see [5], pg. 143) we obtain that m Q 0 0 = 1 for every X e l r r ( C ) . We get b through the Theorem 2.4 of [2]. 2 Remark. 2 The 2-Q^-groups are exactly the 2-Q-groups. Proposition 6. a) Z w r . . . w r Z 2 2 is a Q^-group. b) A 2-group is a Q^4-group i f and only i f it can be embedded without fusion in a direct product of Z wr ... wr Z (wr means wreath product). 2 2 For the proof see [1]. Proposition 7. Let G be a Q-4-group with nonabelian dihedral resp. quaternionic 2-Sylow groups. Then the 2-Sylow groups are isomorphic with D resp. Q . B Proof. D resp. Q are the only dihedral resp. quaternionic groups whose characters are rational valued. 8 8 & nonabelian R E F E R E N C E S [11 A L E X A N D R U , V. and ARMEANU, I. Sur les caractères d'un groupe fini, C R . Acad. Sei. Paris, 298 (1984), Serie I, No. 6. [2] GOW, R. Real-Valued and 2-Rational Group Characters, J. of Algebra, 61 (2) (1979), 388-413. [3] K L E T Z I N G , D. Structure and Representations of Q-groups, Lecture Notes in Mathematics, Springer-Verlag, 1984. [4] ROSE, J.S. A Course in Group Theory, Cambridge, 1978. [5] Y A M A D A , T. The Schur Subgroup of the Brauer Group, Lecture Notes in Mathematics 397, Springer-Verlag, 1974.  (...truncated)