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Is a semidirect product of groups necessarily a group? (English) Zbl 0782.20053

O. Chein [Trans. Am. Math. Soc. 188, 31-51 (1974; Zbl 0286.20088)] provided a construction of a non-associative Moufang loop \(L\) which is a semidirect product of a nonabelian subgroup by a subgroup of order 2. On the other hand, it is known that a commutative group which is a semidirect product of subgroups must be their direct product. The question raised now is: if \(L\) is a commutative loop which is a semidirect product of subgroups, must it necessarily be a group? The authors indeed produce three construction methods, different from Chein’s method, yielding nonassociative commutative loops which are semidirect products of subgroups. In all these constructions, if \(L\) is Moufang it must be a group.
Reviewer: R.Artzy (Haifa)

MSC:

20N05 Loops, quasigroups
20E22 Extensions, wreath products, and other compositions of groups

Citations:

Zbl 0286.20088
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References:

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