2 research outputs found
Automorphism tower problem and semigroup of endomorphisms for free Burnside groups
We have proved that the group of all inner automorphisms of the free Burnside
group is the unique normal subgroup in among all its
subgroups, which are isomorphic to free Burnside group of some rank
for all odd and . It follows that the group of
automorphisms of the free Burnside group is complete for
odd , that is it has a trivial center and any automorphism of
is inner. Thus, for groups is solved the automorphism
tower problem and is showed that it is as short as the automorphism tower of
the absolutely free groups. Moreover, proved that every automorphism of
is a conjugation by an element of .Comment: 5 page
Splitting automorphisms of prime power orders of free Burnside groups
We prove that if the order of a splitting automorphism of free Burnside
group~ of odd period~ is a prime power, then the automorphism
is inner. Thus, we give an affirmative answer to the question on the
coincidence of splitting and inner automorphisms of free Burnside groups
for automorphisms of orders~ ( is a prime number). This
question was posed in the Kourovka Notebook in 1990 (see 11th ed., Question
11.36.~b)