On Frattini subloops and normalizers of commutative Moufang loops
نویسنده
چکیده
Let L be a commutative Moufang loop (CML) with multiplication group M, and let F(L), F(M) be the Frattini subgroup and Frattini subgroup of L and M respectively. It is proved that F(L) = L if and only if F(M) = 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 F(L) 6= L then L satisfies the normalizer condition and that any divisible subgroup of M is an abelian group and serves as a direct factor for M. Classification: 20N05
منابع مشابه
The commutative Moufang loops with maximum conditions for subloops
It is proved that the maximum condition for subloops in a commutative Moufang loop Q is equivalent with the conditions of finite generating of different subloops of the loop Q and different subgroups of the multiplication group of the loop Q. An analogue equivalence is set for the commutative Moufang ZA-loops. Classification: 20N05
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