Leírás
Abstract:
1. Phillips' problem:
In 1968 Glauberman proved that the Feit-Thompson's Theorem can
be extended to Moufang loops, namely every Moufang loop of odd order
is solvable. Glauberman and Doro studied the structure of Moufang
loops of odd orrder, and it turned out that the multiplication group
of a Moufang loop of odd order with trivial nucleus is a group with
triality.
One of the main problems in area of Moufang loops is the so called
Phillips' problem :
Does there exist a Moufang loop of odd order with trivial nucleus?
We give a negative answer by proving that every Moufang loop of odd
order has nontrivial nucleus.
2. On Doro's conjecture
In 1978 S. Doro in his paper published the following conjecture:
If the nucleus of a Moufang loop is trivial, then the commutant
is a normal subloop.
The following problem had been open for a while, and officially
raised by A. Rajah in 2003:
Is the commutant of a Moufang loop normal in the loop?
First Gagola stated that the answer to this question is
affirmative. Grishkov and Zavarnitsine showed that the answer is in
fact generally negative by constructing two infinite series of Moufang
loops of exponent 3, whose commutant is not a normal subloop.
Their results reopened Doro's conjecvure.
By using transversals belonging to the commutant, we characterize
Moufang loops whose commutant is a normal subloop, i.e. we give
necessary and sufficient conditions in the multiplication group for
this purpose.
Applying these characterizations, by working in the multiplication
group of the loop we prove that in case of Moufang loops with trivial
nucleus the commutant is normal if and only if it is trivial.
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