Moufang loops with nonnormal commutative centre

We construct two infinite series of Moufang loops of exponent 3 whose commutative centre (i. e. the set of elements that commute with all elements of the loop) is not a normal subloop. In particular, we obtain examples of such loops of orders 38 and 311 one of which can be defined as the Moufang triplication of the free Burnside group B(3, 3).

AUTOMORPHISMS OF FREE BURNSIDE GROUPS OF PERIOD 3

We have proved that any automorphism of the free Burnside group $ B(3) $ of period 3 and an arbitrary rank is induced by an automorphism of the free group of the same rank.

THE GROUPS OF AUTOMORPHISMS ARE COMPLETE FOR FREE BURNSIDE GROUPS OF ODD EXPONENTS n ≥ 1003

It is proved that the group of automorphisms Aut (B(m, n)) of the free Burnside group B(m, n) is complete for every odd exponent n ≥ 1003 and for any m > 1, that is, it has a trivial center and any automorphism of Aut (B(m, n)) is inner. Thus, the automorphism tower problem for groups B(m, n) is solved and it is showed that it is as short as the automorphism tower of the absolutely free groups. Moreover, the group of all inner automorphisms Inn (B(m, n)) is the unique normal subgroup in Aut (B(m, n)) among all its subgroups, which are isomorphic to free Burnside group B(s, n) of some rank s.

NORMAL AUTOMORPHISMS OF FREE BURNSIDE GROUPS OF LARGE ODD EXPONENTS

Let [Formula: see text] be a free Burnside group of a sufficiently large odd exponent n with a basis [Formula: see text] of cardinality at least 2. We prove that every normal automorphism of [Formula: see text] is inner. We also prove that a free Burnside group of large odd exponent n can be normally embedded into group of exponent n only as a direct factor.