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On a certain class of G-loops. (Russian) Zbl 0588.20043

A loop \((Q,\cdot)\) is called a \(WIP_ n\)-loop if it satisfies the following identity with respect to an involution I on \(Q: I^ n(x\cdot y)\cdot I^{n+1}x=I^ ny\). The main results of the paper are the following. Proposition 2: If each loop which is isotopic with a \(WIP_ n\)-loop \((Q,\cdot)\) is also a \(WIP_ n\)-loop, then each isotope of \((Q,\cdot)\) is isomorphic to the loop \((Q,\circ)\), where \(x\circ y=R_ d^{-1}(x\cdot R_ dy)\), for some \(d\in Q\). Theorem: Under the same hypotheses for (Q,\(\cdot)\) and the hypothesis that for any \(a\in Q\), \(a^ 2\) belongs to the kernel of the loop \((Q,\cdot)\), this loop is a G- loop.

MSC:

20N05 Loops, quasigroups
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