(3, 𝑞, 𝑟)-generations of Fischer's sporadic group Fi′24

Abstract

Abstract A group G is said to be ( l , m , n ) {(l,m,n)} -generated if it can be generated by two suitable elements x and y such that o ⁢ ( x ) = l {o(x)=l} , o ⁢ ( y ) = m {o(y)=m} and o ⁢ ( x ⁢ y ) = n {o(xy)=n} . In [J. Moori, ( p , q , r ) {(p,q,r)} -generations for the Janko groups J 1 {J_{1}} and J 2 {J_{2}} , Nova J. Algebra Geom. 2 1993, 3, 277–285], J. Moori posed the problem of finding all triples of distinct primes ( p , q , r ) {(p,q,r)} for which a finite non-abelian simple group is ( p , q , r ) {(p,q,r)} -generated. In the present article, we partially answer this question for Fischer’s largest sporadic simple group Fi 24 ′ {\mathrm{Fi}_{24}^{\prime}} by determining all ( 3 , q , r ) {(3,q,r)} -generations, where q and r are prime divisors of | Fi 24 ′ | {\lvert\mathrm{Fi}_{24}^{\prime}\rvert} with 3 < q < r {3<q<r} .