Normal Automorphisms Research Articles

Normal automorphisms of free metabelian Leibniz algebras

Let $\mathfrak{M}$ be a free metabelian Leibniz algebra generating set $% X=\{x_{1},...,x_{n}\}$ over the field $K$ of characteristic $0$. An automorphism $ \phi $ of $\mathfrak{M}$ is said to be normal automorphism if each ideal of $\mathfrak{M}$ is invariant under $ \phi $. In this work, it is proven that every normal automorphism of $\mathfrak{M}$ is an IA-automorphism and the group of normal automorphisms coincides with the group of inner automorphisms.

Fixed points and normal automorphisms of the unit ball of bounded operators on $$\mathbb {C}^n$$

We examine the group of isometries of the open unit ball of a complex Banach space of certain bounded linear operators equipped with the Carath\'eodory metric. Therein we obtain a charactrization of the normal isometries in terms of their special type of fixed points.

A Krull–Remak–Schmidt theorem for fusion systems

We prove that the factorization of a saturated fusion system over a discrete $p$-toral group as a product of indecomposable subsystems is unique up to normal automorphisms of the fusion system and permutations of the factors. In particular, if the fusion

On Normal Automorphisms of Groups

On Normal Automorphisms of Groups

Normal Automorphisms of Free Groups of Infinitely Based Varieties

In the paper, automorphisms are studied for free groups of varieties given by a family of identities in the well-known infinite independent system of identities involving two variables that was constructed by S. I. Adian to solve the finite basis problem in group theory. It is proved that every normal automorphism (i.e., an automorphism that stabilizes any normal subgroup) of noncyclic free groups of these varieties is an inner automorphism.

Normal automorphisms of the metabelian product of free abelian Lie algebras

Let M be the metabelian product of free abelian Lie algebras of finite rank. In this study we prove that every normal automorphism of M is an IA-automorphism and acts identically on M′.

Normal automorphisms of a soluble product of abelian groups

Let G be a soluble product of class n ≥ 2 of nontrivial free abelian groups. We prove that the subgroup of all automorphisms of G identical on the last nonunit commutant G(n), coincides with the subgroup of all inner automorphisms corresponding to the elements of G(n). We also show that the subgroup of all normal automorphisms of G coincides with the subgroup of all inner automorphisms.

Automorphisms of Divisible Rigid Groups

A group G is m-rigid if there exists a normal series of the form G = G1 > G2 > . . . > Gm > Gm+1 = 1 in which every factor Gi/Gi+1 is an Abelian group and is torsion-free as a (right) Z[G/Gi]-module. A rigid group is one that is m-rigid for some m. The specified series is determined by a given rigid group uniquely; so it consists of characteristic subgroups and is called a rigid series; the solvability length of a group is exactly m. A rigid group G is divisible if all Gi/Gi+1 are divisible modules over Z[G/Gi]. The rings Z[G/Gi] satisfy the Ore condition, and Q(G/Gi) denote the corresponding (right) division rings. Thus, for a divisible rigid group G, the factor Gi/Gi+1 can be treated as a (right) vector space over Q(G/Gi). We describe the group of all automorphisms of a divisible rigid group, and then a group of normal automorphisms. An automorphism is normal if it keeps all normal subgroups of the given group fixed.

On normal automorphisms of $n$-periodic products of finite cyclic groups

We prove that each normal automorphism of the $n$-periodic product of cyclic groups of odd order $rge1003$ is inner, whenever $r$ divides $n$.

On outer normal automorphisms of periodic products of groups

The paper proves that, as opposed to free products of groups, for any odd n ≥ 665 there are some groups G1 and G2 with n-periodic product G1*nG2 possessing a normal outer automorphism.

Normal automorphisms of free Burnside groups

We prove that for an arbitrary odd and every automorphism of the free Burnside group that stabilizes every maximal normal subgroup of infinite index is an inner automorphism. For the same values of and , we establish that the subgroup of inner automorphisms of is maximal among the subgroups in which the orders of the elements are bounded by .

Normal automorphisms of relatively hyperbolic groups

An automorphism $\alpha$ of a group $G$ is normal if it fixes every normal subgroup of $G$ setwise. We give an algebraic description of normal automorphisms of relatively hyperbolic groups. In particular, we show that for any such group $G$, $Inn(G)$ has finite index in the subgroup $Aut_n(G)$ of normal automorphisms. If, in addition, $G$ is non-elementary and has no finite non-trivial normal subgroups, then $Aut_n(G)=Inn(G)$. As an application, we show that $Out(G)$ is residually finite for every finitely generated residually finite group $G$ with infinitely many ends.

NORMAL AUTOMORPHISMS OF A FREE METABELIAN NILPOTENT GROUP

AbstractAn automorphism φ of a group G is said to be normal if φ(H) = H for each normal subgroup H of G. These automorphisms form a group containing the group of inner automorphisms. When G is a non-abelian free (or free soluble) group, it is known that these groups of automorphisms coincide, but this is not always true when G is a free metabelian nilpotent group. The aim of this paper is to determine the group of normal automorphisms in this last case.

On the Automorphisms of Some One-Relator Groups

The description of the automorphism group of group ⟨ a, b;[a m ,b n ] = 1⟩ (m,n > 1) in terms of generators and defining relations is given. This result is applied to prove that any normal automorphism of every such group is inner.

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.

Normal automorphisms of free solvable pro-p-groups

An automorphism of a profinite group is called normal if it leaves invariant all (closed) normal subgroups. An automorphism of an abstract group is called p-normal if it leaves invariant each normal subgroup of p-power, where p is prime. An inner automorphism satisfies both of these conditions. Earlier, Romanovskii and Boluts [2] gave a description of normal automorphisms of a free solvable pro-p-group of derived length 2. That description implied, in particular, that the number of normal automorphisms in that group exceeds the number of inner ones. Here we prove that each normal automorphism of a free solvable pro-p-group of derived length≥3 and a p-normal automorphism of an abstract free solvable group of derived length≥2 are inner.

Free products of groups have no outer normal automorphisms

An automorphism of an arbitrary group is called normal if all subgroups of this group are left invariant by it. Lubotski [1] and Lue [2] showed that every normal automorphism of a noncyclic free group is inner. Here we prove that every normal automorphism of a nontrivial free product of groups is inner as well.

Normal automorphisms of a free pro-p-group in the varietyN 2 A

An automorphism of a (profinite) group is called normal if each (closed) normal subgroup is left invariant by it. An automorphism of an abstract group is p-normal if each normal subgroup of p-power, where p is prime, is left invariant. Obviously, the inner automorphism of a group will be normal and p-normal. For some groups, the converse was stated to be likewise true. N. Romanovskii and V. Boluts, for instance, established that for free solvable pro-p-groups of derived length 2, there exist normal automorphisms that are not inner. Let N2 be the variety of nilpotent groups of class 2 and A the variety of Abelian groups. We prove the following results: (1) If p is a prime number distinct from 2, then the normal automorphism of a free pro-p-group of rank ≥2 in N2A is inner (Theorem 1); (2) If p is a prime number distinct from 2, then the p-normal automorphism of an abstract free N2A-group of rank ≥2 is inner (Theorem 2).

Normal automorphisms of free solvable pro-p-groups of derived length 2

Normal automorphisms of free solvable pro-p-groups of derived length 2

Normal automorphisms of discrete groups

Normal automorphisms of discrete groups