Simple Moufang loops and alternative algebras
Let a Moufang loop Q contain a non-unitary subloop, which is a simple loop. Then Q is not embedded into a loop of invertible elements of any alternative algebra.
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arXiv subjects
Nicolae Sandu
Publications and source records attributed to Nicolae Sandu.
Simple Moufang loops and Galois extensions
The paper establishes an one-to-one correspondence between simple Moufang loops and Paige loops constructed over Galois extension over prime field in its algebraic closure. Using this connection it describes fully the family of nonassociative finite simple Moufang loops.It describes the generators, the loop structure of subloops, the automorphism group of nonassociative simple Moufang loops.
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On some criterions of finiteness conditions in commutative Moufang loops
The various finiteness conditions in commutative Moufang loops are characterized using the notions of centralizer of subloops and centralizer of subgroups of its multiplication group.
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On Frattini subloops and normalizers of commutative Moufang loops
Let $L$ be a commutative Moufang loop (CML) with multiplication group $\frak M$, and let $\frak F(L)$, $\frak F(\frak M)$ be the Frattini subgroup and Frattini subgroup of $L$ and $\frak M$ respectively. It is proved that $\frak F(L) = L$ if and only if $\frak F(\frak M) = \frak M$ and is described the structure of this CLM. Constructively it is defined the notion of normalizer for subloops in CML. Using this it is proved that if $\frak F(L) \neq L$ then $L$ satisfies the normalizer condition and that any divisible subgroup of $\frak M$ is an abelian group and serves as a direct factor for $\frak M$.
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About the embedding of Moufang loops in alternative algebras
It is proved that any free Moufang loop can be embedded in a loop of invertible elements of some alternative algebra.
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