D. Gorenstein, Finite Groups. Harper and Row, New York 1968.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to be an orbital regular graph if it is G orbital regular for some G Aut#. A Frobenius group is a transitive permutation group on a set V which is not regular on V , but has the property that the only element of G which has more than one fixed point is the identity. It was shown by Thompson(see[4] and [8] that a finite Frobenius group G has a nilpotent normal subgroup K, called the Frobenius kernel, which is regular on V . Hence G is the semidirect product K:H, where H is the stabilizer of a point of V ; each such subgroup H is called a Frobenius complement for K in G. The following ....

....= G . In other words, #(G) G . Look at Table 4.1. Because #(K) H is a Frobenius group, so H acts semi regularly on #(K) 1 . When 1, we can know that the elements in #(K) 1 have equal order. Consiquently, #(K) of K in Table 4.1 are all elementary abelian groups. By Gorenstein ([4], p.38 and p.339) when is even, K is abelian and H possesses a unique involution which necessarily is contained in Z(H) Thus, we can determine the structure of some Frobenius groups when is even. Lemma 4.2 Let G = K:H be a Frobenius group. The Frobenius complement H = Z 3 ; the ....

[Article contains additional citation context not shown here]

D. Gorenstein, Finite Groups. (Harper and Row, New York, 1968).

....satisfied: a. The p rank of P is two; b. F is normal in P ; c. Every other maximal elementary abelian subgroup of P is not contained in the Frattini subgroup. 5 Proof. First, suppose that the normal p rank is one, so every normal abelian subgroup of P is cyclic. Then, by a result of P. Hall ([8], p.198) we have that p = 2 and P is dihedral or quasi dihedral and the condition that F is contained in the Frattini subgroup is violated, by inspection. Hence, we may let N be a normal elementary abelian subgroup of order p in P . Since F is contained in the Frattini subgroup it follows that ....

D. Gorenstein, Finite Groups, Harper and Row 1968

....to denote those of IF q n . The Projective Linear Group of degree 2 over IF q n (PGL 2 (q n ) is the group (under matrix multiplication) of 2 Theta 2 non singular matrices with entries in IF q n , modulo its center, the group of scalar matrices (i.e. scalar multiples of the identity) [6]. In other words, matrices that differ by a scalar multiple represent the same group element. It is well known that jPGL 2 (q n )j = q n 1)q n (q n Gamma 1) 1) 9 A matrix of PGL 2 (q n ) will be usually written either as 2 4 ff fi ffi 1 3 5 or 2 4 ff fi 1 0 3 5 , with ff; ....

D. Gorenstein. Finite Groups. Harper and Row, New York, NY, 1968.

....this situation actually arises and requires additional attention. A forthcoming paper will deal with those groups in F(p) the socle of which is nonabelian. 1 Preliminary results G will always denote a finite group and p a prime. The notation used is standard and can be found in [12] or [7]. In particular (G) denotes the set of prime divisors of jGj; for any set of primes, 0 is the complement of in the set of all primes; O (G) denotes the largest normal subgroup of G and O(G) O f2g 0 (G) Z(G) is the centre of G and F (G) is the Fitting subgroup of G (i.e. the ....

....isotropic subspace U and put G = fg 2 Sp 4 (q) j g(U) U; det(gj U ) 1g. Then O p (G) is an elementary abelian group of order p 3a , and G=O p (G) SL 2 (q) One verifies directly that G = G 0 and G acts faithfully and p 0 semiregularly on V . The following result is well known (cf. [7] Theorems 3.3.3 and 5.4.10) Lemma 1.5 Let (G; V ) be p 0 semiregular and let r 2 (G)nfpg. Then the r rank of G is one, hence Sylow r subgroups of G are cyclic or quaternion groups. ffi Next we see that it suffices to study indecomposable groups: Lemma 1.6 Let F be an arbitrary field ....

D. Gorenstein, Finite Groups, Harper and Row, 1968.

....The blocks f0; 3; 7g and f1; 2; 3; 5g are base blocks for this design with respect to G. The permutations (0; 4; 7; 9; 3) 1; 5; 8; 2; 6) and (0; 3) 1; 2) 4; 9) 5; 8) 6) 7) generate G. 1. 4 Remark Extensive background material on groups and permutation groups is available in standard references [1, 9, 7, 6, 12]. In addition, numerous algebraic computation packages provide support for group theoretic investigations. Pythia, developed by .1.2 Permutation Groups 3 Chouinard and Magliveras, provides a suite of computational group theory programs, and a data base of groups. Cayley [2] and its successor, ....

....D5 10 [1,2 2 ] x 1; Gammax Z 2 3, 4 . 3 AGL1(5) 20 2t x 1;2x Z 4 5 :4 A5 60 3p a3;a5 A4 5 :5 S5 120 5p a4;a5 S4 6:1 L2(5) A5 60 2p a5; b3 5:2 2, 3 :2 PGL2(5) S5 120 3t a5; b6 5:3 4 :3 A6 360 4p a3;a5 5:4 4 :4 S6 720 6p a5;a6 5:5 7:1 Z 7 7 [1 7 ] a7 : x 1 1 2, 3 :2 D7 14 [1,2 3 ] x 1; Gammax Z 2 4 :3 F7;3 21 2 x 1;2x Z 3 4, 5 :4 AGL1(7) 42 2t x 1;3x Z 6 7 :5 L3(2) 168 2t a7; b7 S4 6 S1(2; 3; 7) 6 A7 2520 5p a3;a7 6:3 7 :7 S7 5040 7p a6;a7 6:4 8:1 AGL1(8) 56 2p a7; b7 7:1 2 :2 V8 Delta ....

[Article contains additional citation context not shown here]

D. Gorenstein, Finite Groups, Harper and Row, 1968.

....an abelian subgroup A with d(A) d(M) for some maximal abelian normal subgroup M of P (k Gamma2) Now observe that by our construction ORBIT SIZES AND CHARACTER DEGREES 331 of C and the proof of [3, Theorem 5.3. 12] we see that C is a critical subgroup of P (k Gamma2) in the sense of [3]; so by the proof of [3, Theorem 5.3.13] we know that Omega 1 (C) Theta G is a p group of class 1 or 2 and of exponent p which contains the abelian subgroup A. Hence we may apply 2.1 to the action of G on V which yields that b 1 6 d(A) Now by the proof of [5, III, Satz 7.11] we have jP ....

D. Gorenstein, Finite Groups, Harper and Row, New York, 1968.

....of such a check character system. The size of the alphabet under consideration is then a power of two. In particular, we get systems for alphabets with 64 letters using non abelian groups which are Sylow 2 subgroups of Sz(8) U 3 (4) and L 3 (4) respectively ( for the notations see for instance [2]) For 256 letters, besides the systems over the (abelian) Sylow 2 subgroups of L 2 (2 ) one can use the greatest normal 2 subgroup O 2 (H) of the stabilizer H in U 5 (2) of a 2 dim isotropic subspace with an element of order 5 of H as fixed point free automorphism. There are given as well ....

....Take for instance P 1 : # 1 (Z(P) and define P i inductively such that P P i 1 = # 1 (Z(P P ) Here # 1 (G) denotes the subgroup of G generated by the elements of order p) Suppose first that # acts fixed point freely on P . Then # acts fixed point freely on each P i P i (cf. [2] p 335) Choose x P such that x is conjugate in P to x . Let i be minimal with x P . Suppose i 0. Then x and so x .Now(xP i 1 ) is conjugate to xP i 1 .Asx 1 x #[P i ,P]#P i 1 we have x xP i 1 . Hence #xP =#xP #.As Aut(#xP i 1 #) is cyclic of order p 1, ....

D. Gorenstein, Finite Groups, Harper and Row (1968).

....that the existence of such an element g is an easy consequence of our result. Finally note that the proof of the Theorem makes use of the classification of the finite simple groups. 1. Reduction to simple groups Throughout this text we use standard notation for finite groups as for example in [G], H] or [A] In particular we use the following: jGj denotes the order of the group G; jGj p denotes the maximal p power dividing jGj for a prime number p; G) is the set of prime divisors of jGj; G 0 = G; G] is the commutator subgroup and Z(G) is the centre of G; for x 2 G and N G ....

D.Gorenstein, Finite Groups, Harper and Row, New York (1968);

No context found.

D. Gorenstein, Finite Groups. Harper and Row, New York 1968.

No context found.

D. Gorenstein, Finite Groups, Harper and Row, New York, 1968.

No context found.

D. Gorenstein, Finite Groups. Harper and Row, New York 1968.

No context found.

D. Gorenstein, Finite groups, Harper and Row 1968

No context found.

D. Gorenstein, Finite Groups. Harper and Row, New York 1968.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC