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Vietnam Journal of Mathematics

Khoa Toán, Khoa Khoa học, Đại học Kashan, Kashan, Iran rtment Toán học, Đại học Tarbiat Modarres, | Vietnam Journal of Mathematics 33 1 2005 1-7 V Í e It ini ai m J o mt r im ai I of MATHEMATICS VAST 2005 nX-Complementary Generations of the Rudvalis Group Ru Ali Reza Ashrafi1 and Ali Iranmanesh2 1 Department of Mathematics Faculty of Science University of Kashan Kashan Iran 2 Department of Mathematics Tarbiat Modarres University P.O.Box 14115-137 Tehran Iran Received March 19 2003 Revised October 17 2004 Abstract Let be a finite group and a conjugacy class of elements of order in . is called complementary generated if for every. there is a such that. In 20 the question of finding all positive integers such that a given non-abelian finite simple group is -complementary generated was posed. In this paper we answer this question for the sporadic group . In fact we prove that for any element order of the sporadic group is -complementary generated if and only if . . . 1. Introduction 3 . . . A 5 7 m 7 m 2 g h g h Ali Reza Ashrafi and Ali Iranmanesh 3 2 22 . 1 2 3 2. . . . Lemma 1.1. If is -complementary generated and for some integer then is -complementary generated. Lemma 1.2. .Let be a 2 -generated. -generated simple group then is Lemma 1.3. Let be a finite simple group and a maximal subgroup of containing a fixed element . Then the number of conjugates of containing is . where . is the permutation character of with action on the -Complementary Generations of the Rudvalis Group 3 conjugates of .In particular . G 1 where 1 2. are representatives of the -conjugacy classes that fuse to the -conjugacy class of Theorem. The Rudvalis group is -complementary generated if and only if 3. 2. -Complementary Generations for . Table 1. .4. . 22. 3. 2 2. 2 29 . 6.2 2 2. .5 212.33. 2.3 28. 2 5.3 2. 3. 25.3. 2.3 22 . 2 5.3 3. 23 .3 3 24 .3.2 26 U3. 23 8 L3 2 2.24 6 -5 . 8 .2 5 .1 2 2 5 2 .6.2 2 12.3 3. 214 .3. 214 .3. 2 6.3 2. 25.3. 3 25. 3 .