7,277 research outputs found
Double automorphisms of graded Lie algebras
We introduce the concept of a double automorphism of an A-graded Lie algebra
L. Roughly, this is an automorphism of L which also induces an automorphism of
the group A. It is clear that the set of all double automorphisms of L forms a
subgroup in Aut(L). In the present paper we prove several nilpotency criteria
for a graded Lie algebra admitting a finite group of double automorphisms. We
also give an application of our results to groups admitting a Frobenius group
of automorphisms.Comment: 13 page
Automorphisms fixing every normal subgroup of a nilpotent-by-abelian group
Among other things, we prove that the group of automorphisms fixing every
normal subgroup of a nilpotent-by-abelian group is nilpotent-by-metabelian. In
particular, the group of automorphisms fixing every normal subgroup of a
metabelian group is soluble of derived length at most 3. An example shows that
this bound cannot be improved.Comment: 6 page
Normal and normally outer automorphisms of free metabelian nilpotent Lie algebras
Let L be the free m-generated metabelian nilpotent of class c Lie algebra
over a field of characteristic 0. An automorphism f of L is called normal if
f(I)=I for every ideal I of the algebra L. Such automorphisms form a normal
subgroup N(L) of Aut(L) containing the group of inner automorphisms. We
describe the group of normal automorphisms of L and the quotient group of
Aut(L) modulo N(L).Comment: to appear in Serdica Mathematical Journa
Twisted conjugacy classes in nilpotent groups
A group is said to have the property if every automorphism has an
infinite number of twisted conjugacy classes. We study the question whether
has the property when is a finitely generated torsion-free
nilpotent group. As a consequence, we show that for every positive integer
, there is a compact nilmanifold of dimension on which every
homeomorphism is isotopic to a fixed point free homeomorphism. As a by-product,
we give a purely group theoretic proof that the free group on two generators
has the property. The property for virtually abelian
and for -nilpotent groups are also discussed.Comment: 22 pages; section 6 has been moved to section 2 and minor
modification has been made on exposition; to be published in Crelle
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