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C-loops: extensions and constructions. (English) Zbl 1129.20043

A loop \((C,\cdot)\) is called a ‘C-loop’ if \(x(y\cdot yz)=(xy\cdot y)z\) for any \(x,y,z\in C\). Given two C-loops \(K\) and \(Q\), we say that a loop \(C\) is the ‘extension’ of \(K\) by \(Q\) if \(K\) is a normal subloop of \(C\) and \(C/K\) is isomorphic to \(Q\).
In this note the authors prove that if \(K\) is an Abelian group and \(Q\) is a C-loop then the ‘nuclear’ extension \(C\) (i.e. with \(K\leq N(C)\)) can be characterized by the existence of a C-factor set fulfilling a suitable condition. Furthermore, they discuss ‘central’ extensions, that is when \(K\) is contained in the center of the C-loop extension \(C\). For C-loops with central squares the C-factor set condition for central extensions is much simpler and this allows to construct the corresponding extension by means of a procedure illustrated in §4 which uses small blocks arising from the underlying Steiner triple system. Using this method for the extensions, the authors discuss for which Abelian group \(K\) and Steiner loop \(Q\) there is a ‘non flexible’ C-loop \(C\) with center \(K\) and \(C/K\) is isomorphic to \(Q\).
Finally they prove that the loops of signed basis elements in the standard real Cayley-Dickson algebras are C-loops.

MSC:

20N05 Loops, quasigroups
17A30 Nonassociative algebras satisfying other identities
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References:

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