On Connected Transversals to Subgroups Whose Order is a Product of Two Primes

Abstract

We consider finite groups which have connected transversals to subgroups whose order is a product of two primes p and q. We investigate those values of p and q for which the group is soluble. We can show that the solubility of the group follows if q = 2 and p ≤ 61, q = 3 and p ≤ 31, q = 5 and p ≤ 11. We then apply our results on loop theory and we show that if the inner mapping group of a finite loop has order pq where p and q are as above then the loop is soluble.

Orbits of Inner Mapping Groups

“…The latter statement was considered in several papers of Niemenmaa, Cs org o and Myllyl a [15,13,3,4]. In these papers solvability was proved for various special values of p and q.…”

Loops of Lie type: a p-adic example

“…Without any great loss of generality, one can assume that G is generated by L := σ(G/H) and that the action of G on the homogeneous space G/H, aH → gaH, is faithful. In this case the enveloping group G decomposes as 6 see [41], [47], [50], [49], [47], [60], [59], [72] 7 see [21], [29], [100] 8 see [25], [26], [42], [43], [77], [78], [79], [80], [82], see also [73], pp. 21-22, [17], [21], [28], [44] [64], [76] 9 Actually, Moufang defined the quasigroups which today have her name, which are quasigroup satisfying any one of the Moufang identities.…”

On Connected Transversals to Nonabelian Subgroups

“…It turned out in [10] that the solvability of G also follows provided that H is finite abelian. In this paper, which is a continuation of [8], we consider the following problem: If |H | = pq, where p and q are prime numbers, does it then follow that G is solvable? In [8] Niemenmaa was able to prove this by using the classification of finite simple groups in the following cases: q = 2 and p ≤ 61, q = 3 and p ≤ 31, q = 5 and p = 11.…”