Research article · 2018 · Full text
International Journal of Statistics and Applied MathematicsSome properties of commuting automorphism
Volume 3(1) · Pages 171–172
Research areasFinite Group Theory ResearchAdvanced Topics in AlgebraAdvanced Differential Equations and Dynamical Systems
Abstract
Author-writtenLet G be a group. Definition of commuting automorphism is given. We give some properties of commuting automorphism
Summary
Source-checkedPlain-language reading of the full paper, prepared with AI and checked against the text. Each point links to the authors' own words.
Why it matters
- A commuting automorphism is a group automorphism where every element in the group commutes with its image under that automorphism. View evidence ↗
- An A(G) group is defined as a group where the set of all its commuting automorphisms constitutes a subgroup of the full automorphism group. View evidence ↗
- For simple non-abelian groups, the set of commuting automorphisms is trivial (A(G) = 1). View evidence ↗
The question
- The research area explores whether A(G) is always a subgroup of Aut(G) and seeks the conditions under which A(G) equals the group of central automorphisms. View evidence ↗
What they did
- The paper provides a compilation of properties and theorems related to commuting automorphisms. View evidence ↗
- The analysis utilizes the notation A(G) to represent the set of all commuting automorphisms of a group G. View evidence ↗
What they found
- If the center of the commutator subgroup Z(G') has no involutions, A(G) is a subgroup of Aut(G) if and only if the commutators of elements in A(G) are central automorphisms. View evidence ↗
- For any group G, if g is a commuting automorphism, then the commutator [G2, g] is a subgroup of the second center Z2(G). View evidence ↗
- For a commuting automorphism g and elements x and y in G, the commutator of the image of x with y is equal to the commutator of x with the image of y. View evidence ↗
- A group of nilpotency class 2 is an A(G) group if the rank of G/Z(G) is 2. View evidence ↗
- The commutator [f(x), x] equals the identity element {e}, even if x and f(x) are not in the center Z(G). View evidence ↗
- The inverse of the image, (f(x))^-1, commutes with the element x. View evidence ↗
What it means
- The author concludes by presenting a set of properties and theorems that characterize commuting automorphisms and A(G) groups. View evidence ↗
Visual summary
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Commuting Automorphism Concept
AI-generated conceptual illustration (Abstract geometric representation of a mathematical group structure). It explains the idea and does not show study data.
View source evidence ↗Introduction
An automorphism g of G is called a commuting automorphism if [g(x), x] = 1 for all x G, i.e. each element x in G commutes with its image under g [5] . A(G) denotes the set of all commuting automorphisms of G. A group is said to be an A(G) group if the set of all commuting automorphisms of G forms a subgroup of Aut(G). If G is a simple non-abelian group then A(G) = 1 [5,2] , this problem was given by Herstein to the American Mathematical Monthly. For this problem, Laffey proved that A(G) =1, if G has no non-trivial abelian normal subgroups [5,4] . Pettet found that A(G) = 1, If Z(G)=1 and G' = G. Deaconescu, Silberberg and walls gave problems about A(G). The problems were whether A(G) is always a subgroup of Aut(G), to find conditions on G st A(G)=Autz (G). Deaconescu et al. gave an example of a group of order 2 5 s.t. A(G) does not form a subgroup of Aut(G). Vosooghpour, Malayeri proved that for G to be non-A(G) p-group the minimum order of G is p 5 . Recently P.K. Rai gave sufficient conditions on a finite p-group G s.t. G is an A(G) -group. [6] An automorphism of G is called a central automorphism If it commutes with all inner automorphisms or equivalently, if g
Basic facts 2.1
A cyclic group is generated by a single element. We denote a cyclic group of order n by Cn. The rank of a group is denoted by d(G), which is the smallest generating set of G. The least common multiple of orders of the elements of a finite group G is called the exponent of G.
is exactly the set of all non-generators of G. Let G be a finite group then G is nilpotent if and only of G/(G) is nilpotent. [3] A finite collection of normal subgroups Hi of a group G is a normal series for G if 1 = H0 H1 H2 ....... Hr = G. This normal series is a central series if Hi/Hi-1 Z (G/Hi-1) for 1 < i < r. A group G is nilpotent if it has a central series. Subgroups and factor groups of nilpotent groups are nilpotent. Given any group G, we define a central series as follows. Let Z0 = 1 and Z1 = Z (G). The second center Z2 is defined to be the unique subgroup such that Z2/Z1 = Z (G/Z1). We continue like this, inductively defining Zn for n > 0 so that Zn/Zn-1 = Z (G/Zn-1). The chain of normal subgroups.
Constructed in this way is called upper central series of G. The upper central series may not actually be a central series for G because it may happen that Zi < G for all i. If Zr = G for some integer r, then {Zi |0 < i < r} is a true central series and G is nilpotent. If G is an arbitrary nilpotent group, then G is a term of its upper central series. As G = Zr for some integer r > 0, and the smallest integer r for which this happens is called the nilpotence class of G. Non trivial abelian groups have nilpotence class 1, and for non-abelian groups of nilpotency class 2 have quotient group G/Z(G) abelian.
2.2 Lemma: ( [3] ) Let G be finite. Then the following are equivalent. 1. G is nilpotent 2. Every nontrivial homomorphic image of G has a nontrivial center. 3. G appears as a member of its upper central series.
2.3 Theorem. ( [3] ) Let G be a (not necessarily finite) nilpotent group with central series. 1 = H0 H1 H2 ....... Hr = G, and let 1 = Z0 Z1 Z2 ....... be the upper central series of G. Then Hi Zi for 0 < i < r, and in particular, Zr = G. Theorem 2.4 ( [3] ) Let H is a subgroup of G, where G is a nilpotent group. Then NG(H) > H. 3. Some Results on commuting automorphisms. Theorem 3.1 ( [5,1] ) Let G be a group s.t. Z(G') contains no involutions. Then A(G) is a subgroup of Aut(G) Iff commutators of elements in A(G) are central automorphisms Theorem 3.2 ( [5,1] ) If G is a group and g A(G) then [G 2 , g] < Z2 (G) Theorem 3.3 ( [5,4] ) If g A(G) and x, y G then [g(x), y] = [x, g(y)] Theorem 3.4 ( [5,7] ) Let G be a group of nilpotency class 2. If d(G/Z(G)) = 2 then G is an A(G) group. Theorem 3.5 ( [5,7] ) For a given prime p, the minimal number of generators of a non-A(G) p-group of order p 5 and of nilpotency class 2 is equal to 1.
References
6 in the original paper- Deaconescu M, Silberberg, Walls GL. On commuting automorphisms of groups, Arch. Math. (Basel). 2002; 79(6):423-429.
- Herstein IN. Problems and Solutions: Elementary Problems:E3039, Amer. Math. Monthly, 1984; 91(3):203.
- Laffey TJ. Problems and Solutions: Solutions of Elementary Problems: E3039, Amer. Math. Monthly, 1986; 93(10):816-817.
- Rai, P.K. On commuting automorphisms of finite pgroups, Proc. Japan Acad., 91, Ser. A (2015).
- Singh, S. On automorphism Groups of Finitely Generated Groups, Ph.D. Thesis Thapar University, 2015.
- Vosooghpour, F. and Akhavan-Malayeri, M. On Commuting automorphisms of p-groups, Comm. Algebra, 41(2013), no. 4, 1292-1299.