Automorphisms of n-Ary Groups (original) (raw)

On autotopies and automorphisms of n-ary linear quasigroups

2004

In this article we study structure of autotopies, automorphisms, autotopy groups and automorphism groups of n- ary linear quasigroups. We find a connection between automorphism groups of some special kinds of n-ary quasigroups (idempotent quasigroups, loops) and some isotopes of these quasigroups. In binary case we find more detailed connections between automorphism group of a loop and automorphism group of some its isotope. We prove that every finite medial n-ary quasigroup of order greater than 2 has a non- identity automorphism group. We apply obtained results to give some information on auto- morphism groups of n-ary quasigroups that correspond to the ISSN code, the EAN code and the UPC code.

On Equality of Certain Automorphism Groups

Communications in Algebra, 2016

Let G = H ×A be a group, where H is a purely non-abelian subgroup of G and A is a non-trivial abelian factor of G. Then, for n ≥ 2, we show that there exists an isomorphism φ : Aut γn(G) Z(G) (G) → Aut γn(H) Z(H) (H) such that φ(Aut n−1 c (G)) = Aut n−1 c (H). Also, for a finite non-abelian p-group G satisfying a certain natural hypothesis, we give some necessary and sufficient conditions for Autcent(G) = Aut n−1 c (G). Furthermore, for a finite non-abelian p-group G we study the equality of Autcent(G) with Aut γn(G) Z(G) (G).

ON THE COINCIDENCE OF AUTOMORPHISM GROUPS OF FINITE P -GROUPS

Quaestiones Mathematicae, 2023

Let G be a finite non-abelian p-group, where p is a prime. Let γn(G) and Zn(G) respectively denote the nth term of the lower and upper central series of G, and let Ln(G) denote the nth absolute center of G. An automorphism α of G is called an IA n-automorphism if x −1 α(x) ∈ γn+1(G) for all x ∈ G. The group of all IA n-automorphisms of G is denoted by IA n (G). Let IA n Zn(G) (G) denote the subgroup of IA n (G) which fixes Zn(G) elementwise. In this paper, we give necessary and sufficient conditions for a finite non-abelian p-group G of class (n + 1) such that IA n Zn(G) (G) = Aut M Ln(G) (G), where M is a subgroup of G such that γn+1(G) ≤ M ≤ Z(G) ∩ Ln(G), and as a consequence, we obtain the main result of Chahal, Gumber and Kalra [7, Theorem 3.1]. We also give necessary and sufficient conditions for a finite non-abelian p-group G of class (n+1) such that Autz(G) = IA n (G), and obtain Theorem B of Attar [4] as a particular case.

Remarks to Glazek's results on n-ary groups

2007

It is a survey of the results obtained by K. Glazek's and his co-workers. We restrict our attention to the problems of axiomatizations of n-ary groups, classes of n-ary groups, properties of skew elements and homomorphisms induced by skew elements, constructions of covering groups, classifications and representations of n-ary groups. Some new results are added too.

Discussion on Group of Inner Automorphisms of Some Groups

International Journal of Mathematics Trends and Technology, 2017

Let G be an arbitrary group with number of centralizers & is any finite number. In this article, we have proved that the group of inner automorphisms of G is isomorphic to some other groups depending upon . Moreover if for some group , the group of inner automorphisms has order 6 or 9 then will be a group with 5 centralizers and if for some group the group of inner automorphisms has order 4 then will be a group with 4 centralizers & conversely. Notations: (i) -groups: The groups with number of centralizers. (ii) Inn(G): The group of inner automorphisms of any group G. Introduction: All groups mentioned in this paper are finite group. Thus, one expects group structure to become increasingly complex with decreasing 'abelianness.' Indeed, the basic classification scheme for groups reflects the importance of the notion of commutativity. Beginning abstract algebra students tend to ignore the subtleties of the commutativity issue xy = yx as far as they are concerned. An effective ...

Automorphisms of Some 𝒑 − Groups of Order 𝒑

Zenodo (CERN European Organization for Nuclear Research), 2022

There are fifteen groups of order , Out of which five are abelian and the rest are non-abelian. In this paper, we compute the automorphisms of some non-abelian groups of order , where is an odd prime and the verification of number of automorphisms has been made through GAP (Group Algorithm Programming) software.

Remarks to Głazek's results on n-ary groups

Discussiones Mathematicae - General Algebra and Applications, 2007

It is a survey of the results obtained by K. G lazek's and his coworkers. We restrict our attention to the problems of axiomatizations of n-ary groups, classes of n-ary groups, properties of skew elements and homomorphisms induced by skew elements, constructions of covering groups, classifications and representations of n-ary groups. Some new results are added too.