Random Generators of Given Orders and the Smallest Simple Moufang Loop

The probability that m randomly chosen elements of a finite power associative loop C have prescribed orders and generate C is calculated in terms of certain constants Γ related to the action of Aut(C) on the subloop lattice of C. As an illustration, all meaningful probabilities of random generation by elements of given orders are found for the smallest nonassociative simple Moufang loop. 1. Random generators of given orders Let C be a power associative loop, i.e., a loop where every element generates a group. Given an m-tuple a = (a0, . . . , am−1) of elements of C, let A = {Ai} m i=0 be the sequence of nested subloops Ai ≤ C such that A0 is the smallest subloop of C, and Ai+1 = 〈Ai, ai〉. Note that Am is independent of the order of the elements a0, . . . , am−1 in a. Denote by Genm(C) the set of all m-tuples a ∈ C m with Am = C. Then the probability that m randomly chosen elements of C generate C is pm(C) = |C| −m · |Genm(C)|. (1) This notion can be refined in a natural way. For 1 ≤ i ≤ n, let Di be the set of all elements of C of order i. Two m-tuples of integers r = (r0, . . . , rm−1), s = (s0, . . . , sm−1) are said to be of the same type if r0, . . . , rm−1 is a permutation of s0, . . . , sm−1. We say that a = (a0, . . . , am−1) ∈ C m is of type r if there is s = (s0, . . . , sm−1) of the same type as r satisfying ai ∈ Dsi , for 0 ≤ i ≤ m − 1. Let Genr(C) ⊆ Genm(C) be the set of generating m-tuples of type r. Then pr(C) = |C| −m · |Genr(C)| (2) is the probability that m randomly chosen elements a0, . . . , am−1 ∈ C generate C and (a0, . . . , am−1) is of type r. For A, B ≤ C and an integer i, let Γi(A, B) be the cardinality of the set of all elements x ∈ Di such that 〈A, x〉 ∈ OB, where OB is the orbit of B under the natural action of Aut(C) on the subloop lattice of C. Also, let Γ(A, B) be the cardinality of the set of all elements x ∈ C such that 〈A, x〉 ∈ OB. 1991 Mathematics Subject Classification: Primary: 20N05, Secondary: 20F05, 06B99.