### Applied and Computational Mathematics

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### Some Triple Generations of the Lyons Sporadic Simple Group Ly

According to the classification theorem, the Lyons group Ly is one of the 26 sporadic simple groups and has order 51765179004000000 = 28•37•56•7•11•31•37•67. Since the completion of the classification of all finite simple groups, attention has now turned to other aspects e.g. generations of finite groups which entails determining elements which generate that finite group. As a finite nonabelian simple group, Ly can be generated by a minimum of two of its elements. We thus endeavour in the current study to determine some of the pairs of its elements of distinct prime orders from disctinct conjugacy classes with their product in another conjugacy class of elements of prime order which generate Ly and we call such generations triple generations. Triple generations of any finite group are used in the study of its symmetric genus, where the symmetric genus of a Hurwitz group G, of which Ly is known to be a Hurwitz group, is given by . If G is a finite group and lX, mY, nZ are conjugacy classes of elements of G, then G is said to be (l,m,n)-generated if with o(x) = l, o(y) = m and o(xy) = n. The number of distinct ordered pairs (x,y) satisfying such that xy = z, where is an arbitrary class representative, is denoted by ζG(lX,mY,nZ) and is known as the structure constant of the group algebra . The structure constants can be computed from the ordinary character table of G. We shall use the method of the structure constants to determine such generation and/or nongeneration. Thus the object in this paper is to study some of the triple generations of Ly which will thus pave the way towards the study of various combinations of three, four, five etc elements from distinct conjugacy classes which can generate Ly and lead to the ultimate determination of the maximum number of elements of Ly from distinct conjugacy classes of its elements which can generate Ly.

(p,q,r)-Generations, Maximal Subgroups, Primes, Structure Constants, Conjugacy Class Fusions

Malebogo Motalane, Zwelethemba Mpono. (2023). Some Triple Generations of the Lyons Sporadic Simple Group Ly. Applied and Computational Mathematics, 12(3), 55-81. https://doi.org/10.11648/j.acm.20231203.12