AUTOMORPHISM GROUPS OF ENDOMORPHISM MONOIDS OF FREE G-SETS

2014 ◽  
Vol 07 (01) ◽  
pp. 1450015 ◽  
Author(s):  
Janusz Konieczny
Keyword(s):  
Free Algebra ◽  
Automorphism Groups ◽  
Right Action ◽  
Group Of Automorphisms ◽  
Endomorphism Monoids ◽  
Group A

Let G be a group. A G-set is a nonempty set A together with a (right) action of G on A. The class of G-sets, viewed as unary algebras, is a variety. For a set X, let AG(X) be the free algebra on X in the variety of G-sets. We determine the group of automorphisms of End (AG(X)), the monoid of endomorphisms of AG(X).

Nilpotent Partition-Inducing Automorphism Groups

1981 ◽  
Vol 33 (2) ◽  
pp. 412-420 ◽  
Author(s):  
Martin R. Pettet
Keyword(s):  
Finite Group ◽  
Automorphism Groups ◽  
Group Of Automorphisms ◽  
A Finite Group ◽  
Identity Elements

If A is a group acting on a set X and x ∈ X, we denote the stabilizer of x in A by CA(x) and let Γ(x) be the set of elements of X fixed by CA(x). We shall say the action of A is partitive if the distinct subsets Γ(x), x ∈ X, partition X. A special example of this phenomenon is the case of a semiregular action (when CA (x) = 1 for every x ∈ X so the induced partition is a trivial one). Our concern here is with the case that A is a group of automorphisms of a finite group G and X = G#, the set of non-identity elements of G. We shall prove that if A is nilpotent, then except in a very restricted situation, partitivity implies semiregularity.

Centraliser near-rings determined by fixed point free automorphism groups

1987 ◽  
Vol 107 (3-4) ◽  
pp. 327-337 ◽  
Author(s):  
Peter Fuchs ◽  
C. J. Maxson ◽  
M. R. Pettet ◽  
K. C. Smith
Keyword(s):  
Fixed Point ◽  
Free Group ◽  
Automorphism Groups ◽  
Group Of Automorphisms ◽  
Nontrivial Ideal

SynopsisLet G be a group and let A be a fixed point free group of automorphisms of G. It is shown that the centraliser near-ring MA(G) has at most one nontrivial ideal. Conditions on the pair (A, G) are given which force MA(G) to be simple. It is shown that if a nonsimple near-ring MA(G) exists, then A and G have unusual properties.

Automorphism Groups of the Imprimitive Complex Reflection Groups II

2011 ◽  
Vol 18 (02) ◽  
pp. 315-326
Author(s):  
Li Wang
Keyword(s):  
Automorphism Group ◽  
Normal Subgroup ◽  
Automorphism Groups ◽  
Reflection Group ◽  
Reflection Groups ◽  
Group Of Automorphisms ◽  
Complex Reflection ◽  
Complex Reflection Groups ◽  
Complex Reflection Group ◽  
Scalar Multiple

We prove that the automorphism group Aut (m,p,n) of an imprimitive complex reflection group G(m,p,n) is the product of a normal subgroup T(m,p,n) by a subgroup R(m,p,n), where R(m,p,n) is the group of automorphisms that preserve reflections and T(m,p,n) consists of automorphisms that map every element of G(m,p,n) to a scalar multiple of itself.

On the structure and some numerical properties of subgroups and factor-groups defined by automorphism groups

2015 ◽  
Vol 14 (05) ◽  
pp. 1550075
Author(s):  
Leonid A. Kurdachenko ◽  
Javier Otal ◽  
Alexander A. Pypka
Keyword(s):  
Commutator Subgroup ◽  
Automorphism Groups ◽  
Factor Group ◽  
Group Of Automorphisms ◽  
Special Rank ◽  
The Relationship

A group of automorphisms A of a group G defines the A-center CG(A) of G and the A-commutator subgroup [G, A] of G that naturally extend the ordinary center and the commutator subgroup of G. In this paper we study the relationship between the factor group G/CG(A) and the subgroup [G, A] when A has finite special rank and Inn (G) ≤ A.

scholarly journals Cubic Edge-Transitive Bi-$p$-Metacirculants

10.37236/6417 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Yan-Li Qin ◽  
Jin-Xin Zhou
Keyword(s):  
Automorphism Group ◽  
Cayley Graph ◽  
Cayley Graphs ◽  
Infinite Family ◽  
Group Of Automorphisms ◽  
Group A ◽  
Semisymmetric Graphs ◽  
Edge Transitive

A graph is said to be a bi-Cayley graph over a group $H$ if it admits $H$ as a group of automorphisms acting semiregularly on its vertices with two orbits. For a prime $p$, we call a bi-Cayley graph over a metacyclic $p$-group a bi-$p$-metacirculant. In this paper, the automorphism group of a connected cubic edge-transitive bi-$p$-metacirculant is characterized for an odd prime $p$, and the result reveals that a connected cubic edge-transitive bi-$p$-metacirculant exists only when $p=3$. Using this, a classification is given of connected cubic edge-transitive bi-Cayley graphs over an inner-abelian metacyclic $3$-group. As a result, we construct the first known infinite family of cubic semisymmetric graphs of order twice a $3$-power.

scholarly journals Automorphism groups of complex doubles of Klein surfaces

1994 ◽  
Vol 36 (3) ◽  
pp. 313-330 ◽  
Author(s):  
E. Bujalance ◽  
A. F. Costa ◽  
G. Gromadzki ◽  
D. Singerman
Keyword(s):  
Riemann Surface ◽  
Compact Riemann Surface ◽  
Automorphism Groups ◽  
Natural Question ◽  
Klein Surface ◽  
Klein Surfaces ◽  
Maximal Symmetry ◽  
Group Of Automorphisms

In this paper we consider complex doubles of compact Klein surfaces that have large automorphism groups. It is known that a bordered Klein surface of algebraic genus g > 2 has at most 12(g − 1) automorphisms. Surfaces for which this bound is sharp are said to have maximal symmetry. The complex double of such a surface X is a compact Riemann surface X+ of genus g and it is easy to see that if G is the group of automorphisms of X then C2 × G is a group of automorphisms of X+. A natural question is whether X+ can have a group that strictly contains C2 × G. In [8] C. L. May claimed the following interesting result: there is a unique Klein surface X with maximal symmetry for which Aut X+ properly contains C2 × Aut X (where Aut X+ denotes the group of conformal and anticonformal automorphisms of X+).

scholarly journals Some automorphisms of finite nilpotent groups

1960 ◽  
Vol 4 (4) ◽  
pp. 204-207 ◽  
Author(s):  
J. C. Howarth
Keyword(s):  
Finite Groups ◽  
Lower Central Series ◽  
Nilpotent Groups ◽  
Central Series ◽  
Endomorphism Semigroup ◽  
Group Of Automorphisms ◽  
Inner Automorphism ◽  
Group A ◽  
Special Case

This note extends the concept of the inner automorphism, but here applies only to those finite groups G for which some member of the lower central series is Abelian. In general (e.g. when G is metabelian) the construction yields an endomorphism semigroup, but in the special case where Gis nilpotent (and may therefore, for our present purposes, be considered as a p-group) a group of automorphisms results.

scholarly journals On the order of automorphism groups of Klein surfaces

1985 ◽  
Vol 26 (1) ◽  
pp. 75-81 ◽  
Author(s):  
J. J. Etayo Gordejuela
Keyword(s):  
General Result ◽  
Special Interest ◽  
Riemann Surfaces ◽  
Prime Order ◽  
Automorphism Groups ◽  
Klein Surface ◽  
Klein Surfaces ◽  
Group Of Automorphisms

A problem of special interest in the study of automorphism groups of surfaces are the bounds of the orders of the groups as a function of the genus of the surface.May has proved that a Klein surface with boundary of algebraic genus p has at most 12(p–1) automorphisms [9].In this paper we study the highest possible prime order for a group of automorphisms of a Klein surface. This problem was solved for Riemann surfaces by Moore in [10]. We shall use his results for studying the Klein surfaces that are not Riemann surfaces. The more general result that we obtain is the following: if X is a Klein surface of algebraic genus p, and G is a group of automorphisms of X, of prime order n, then n ≤ p + 1.

S-homogeneity and automorphism groups

1993 ◽  
Vol 58 (4) ◽  
pp. 1302-1322
Author(s):  
Elisabeth Bouscaren ◽  
Michael C. Laskowski
Keyword(s):  
Closed Subgroup ◽  
Automorphism Groups ◽  
Single Point ◽  
Stable Structure ◽  
Group Of Automorphisms ◽  
Negative Results ◽  
Weaker Conditions ◽  
Stable Structures

AbstractWe consider the question of when, given a subset A of M, the setwise stabilizer of the group of automorphisms induces a closed subgroup on Sym(A). We define s-homogeneity to be the analogue of homogeneity relative to strong embeddings and show that any subset of a countable, s-homogeneous, ω-stable structure induces a closed subgroup and contrast this with a number of negative results. We also show that for ω-stable structures s-homogeneity is preserved under naming countably many constants, but under slightly weaker conditions it can be lost by naming a single point.

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