On (2, 3, t)-Generations for the Rudvalis Group Ru

A group G is said to be -generated if it can be generated by an involution x and an element y so that 2,3, t    3 o y  and . In the present article, we determine all   o xy t    2,3, t -generations for the Rudvalis sporadic simple group Ru, where t is any divisor of Ru .


Introduction
A group G is said to be -generated if it can be generated by two of its elements x and y so that , and .It is well known that every finite simple group can be generated by just two of its elements.Since the classification of all finite simple groups, more recent work in group theory has involved the study of internal structure of these group and generation type problems have played an important role in these studies.Recently, there has been considerable amount of interest in such type of generations.A -generated group is a homomorphic image of the projective special linear group .It has been known since 1901 (see [1]) that the alternating groups A n are  -generated.Macbeath [2] proved that projective special linear groups ,  , , 2, q 9 q  are -generated.With the exception of Matheiu groups M 11 , M 22 , M 23 , and Maclarin's group McL, all sporadic simple groups are  -generated (Woldar [3]).Gural- nick showed that any non-abelian finite simple group can be generated by an involution and a Sylow 2-subgroup.In addition, a large number of Lie groups and classical linear groups are  -generated as well.Recently, Liebeck and Shalev proved that all finite classical groups (with some exceptions) are -generated.
We say that a group G is   2,3,t -generated (or -generated) if it can be generated by just two of its elements x and y such that x is an involution, 2,3 Fi .Further, Ganief and Moori determined the   2,3,t -generations for the Janko's third sporadic simple group J 3 (see [5]).Recently, the author with others computed   2,3,t -generations for the Held's sporadic simple group He, Tits simple group Conway's two sporadic simple groups Co 3 and Co 2 (see [6][7][8]).Darafsheh and Ashrafi [9] computed generating pairs for the sporadic group Ru.In the present article, we compute all the   t 2,3, -generations for the Rudvalis simple group Ru, where t is any divisor of Ru .

Preliminaries
In this article, we use same notation as in [6].In particular, for C 1 , C 2 and C 3 conjugacy classes of elements the group Ru and g 3 is a fixed representative of C 3 , we define . We can compute the structure of G, where 1 2 , , , m     are the irreducible complex characters of the group G.Further let, denotes the number of distinct tuples   where is obtained by summing the structure constants A general conjugacy class of elements of order n in G is denoted by nX.For examples, 2A represents the first conjugacy class of involutions in a group G.Most of the time, it will clear from the context to which conjugacy classes lX, mY and nZ we are referring.In such case, we suppress the conjugacy classes, using and Then

Main Results
The Rudvalis group Ru is a sporadic simple group of order  1) as also listed in the of Finite Group (see [12]).It has precisely two classes of involution, namely 2A and 2B and a unique class 3A of elements of order 3 in Ru.


It is a well known that if G is -generated finite simple group, then 2,3,t . It follows that we need to consider the cases when   . Further, since a fixed element  in Ru is contained in a two conjugates of the maximal subgroup H 3 , the total contribution from the maximal subgroup H 3 to the structure constant  

H H H A A A A A A A A A A A A
Hence, the group Ru is  2 ,3 ,15  -generated.Next, consider the case .We compute the algebra structure constant as .From the maximal subgroups of Ru, we see that the maximal subgroups that may contain -generated proper subgroups are isomorphic to H 3 , H 9 , H 11 and H 15 .By considering the fusion maps from the these maximal subgroups into the group Ru and the values of h which we compute using Theorem 1, we obtain 2 ,3 ,15 0 2 ,3 ,15 ,15 510 2 45 4 30 > 0.
Therefore,   Case : From the list of maximal subgroups of Ru (Table 1) we observe that, up to isomorphism, H 4 and H 6 are the only maximal subgroups that admit -generated subgroups.From the structure constant we calculate , 2 ,3 ,8 . Since a fixed element in Ru is contained in three conjugate copies of subgroup H 6 , we have , and therefore Ru is -generated.

Case
: For this triple we calculate the structure constant .Up to isomorphism, H 4 , H 6 and H 14 are the only maximal subgroups subgroups of Ru that meet the conjugacy classes 2B, 3A and 8C.We compute that and

>0
. A fixed element a fixed element of order 8 in Ru-class 8C is contained in eight copies of the subgroup H 14 .We obtain , showing that is a generating triple of the group Ru.

Ru
In order to investigate these triples, we construct the group Ru explicitly by using its standard generators given by Wilson [14].
A A Hence by Lemma 2.1, we obtain , proving that Ru is not generated by the triple Since a fixed element of order 10 is contained in two conjugate copies of H 1 , four conjugate copies of H 7 and a unique conjugate copy of H 6 .Therefore proving the generation of Ru by the triple   . Thus, we have and the generation of Ru by this triple follows.
Next, we consider the triple   2 or this triple, the maximal subgroups that meet the Ru classes 2B, 3A and 14Z, up to isomorphism, are H 3 , H 4 , H 9 and H 13 .
Our computation shows that   roving that   .We a proce ous p en in Conder [15] for CAYLEY), in the computer algebra system  (see [16]

H H H A A A A A A A A A
] completely determined the maximal subgroups of the group Ru.It has exactly 15 conjugacy classes of maximal subgroups (see Table Ru class 2A does not meet the maximal subgroup H 9 .The fusion map of the maximal subgroups H 3 into the group Ru yields A. From the above list of maximal subgroups, where 2a, 3a, 3b, 15a, 15b and 2A, 3A, 15A are conjugacy classes of elements in the groups H 3 and Ru, respectively.With the help of this fusion map, we calculate the structure   3 2 ,3 ,15 15

Table 1 . Maximal subgroups of rudvalis group Ru.
Similarly by considering the fusion maps from the maximal subgroups H 8 , H 11 and H 15 we compute that , 8 H  is 2 × 15.
The Rudavalis group Ru is 2 ,3 ,8  We will investigate each triple separately.
H 11 , H 14 and H 15 are the only maximal subgroups of Ru that meet the classes in this triple.We calculate

Table 1 ,
the only maximal subgroups of Ru that meet the classes 2A, 3A and 14Z are isomorphic to H 3 , H 4 and H 13 .Further, H 4 is the only maximal subgroup that contribute to the structure constant as H 10 and H 15 (see Table1).We now consider each case separately.
  