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On finite loops whose inner mapping groups have small orders. (English) Zbl 0881.20006

Let \(G\) be a finite group, \(H\) a subgroup of \(G\) of the order \(pq\) where \(p>q\) are different primes and let \(A\) and \(B\) be two left transversals to \(H\) in \(G\) such that the commutator subgroup \([A,B]\) is a subgroup of \(H\) and \(G\) is generated by \(A\) and \(B\). Then \(G\) is soluble at least in the following cases: (a) \(q\) is not a factor of \(p-1\); (b) \(q=2\) and \(p=3,5,7\); (c) \(q=3\) and \(p=7\) (Theorem 1). If \(Q\) is a loop then the left and right translations are permutations of \(Q\) and they generate the multiplication group \(M(Q)\) of the loop \(Q\). The stabilizer of the neutral element of \(Q\) is denoted by \(I(Q)\) and is called the inner mapping group of \(Q\). If \(Q\) is a finite loop such that \(|I(Q)|=pq\) where \(p\) and \(q\) are two different primes as in Theorem 1 then \(Q\) is a soluble group.
Reviewer: L.Bican (Praha)

MSC:

20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20N05 Loops, quasigroups
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
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