Autocommutators and auto-Bell groups
2
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
Let x be an element of a group G and be an automorphism of G. Then for a positive integer n, the autocommutator [x,n α] is defined inductively by [x, α] = x−1xα = x−1α(x) and [x,n+1 α] = [[x,n α], α]. We call the group G to be n-auto-Engel if [x,n α] = [α,n x] = 1 for all x ∈ G and every α ∈ Aut(G), where [α, x] = [x, α]−1. Also, for any integer n 6= 0, 1, a group G is called an n-auto-Bell group when [xn, α] = [x,αn] for every x ∈ G and each α ∈ Aut(G). In this paper, we investigate the properties of such groups and show that if G is an n-auto-Bell group, then the factor group G/L3(G) has finite exponent dividing 2n(n − 1), where L3(G) is the third term of the upper autocentral series of G. Also, we give some examples and results about n-auto-Bell abelian groups.
Abstract
Let x be an element of a group G and be an automorphism of G. Then for a positive integer n, the autocommutator [x,n α] is defined inductively by [x, α] = x−1xα = x−1α(x) and [x,n+1 α] = [[x,n α], α]. We call the group G to be n-auto-Engel if [x,n α] = [α,n x] = 1 for all x ∈ G and every α ∈ Aut(G), where [α, x] = [x, α]−1. Also, for any integer n 6= 0, 1, a group G is called an n-auto-Bell group when [xn, α] = [x,αn] for every x ∈ G and each α ∈ Aut(G). In this paper, we investigate the properties of such groups and show that if G is an n-auto-Bell group, then the factor group G/L3(G) has finite exponent dividing 2n(n − 1), where L3(G) is the third term of the upper autocentral series of G. Also, we give some examples and results about n-auto-Bell abelian groups.
참고문헌 (13)
- R. Brandl, "On n-Bell groups", Comm. Algebra, 17(4), pp.787~807 (1989)
- M. Deaconescu, "On commuting automorphisms of groups", Arch. Math. (Basel), 79(6), pp.423~429 (2002)
- M. Deaconescu, "Cyclic groups as autocommutator groups", Comm. Al-gebra, 35(1), pp.215~219 (2007)
- C. Delizia, "The structure of Bell groups", J. Group Theory, 9(1), pp.117~125 (2006)
- "GAP-Groups, Algorithms and Programming, Version 4.4.12" (2008)[URL]
- P. V. Hegarty, "The absolute centre of a group", J. Algebra, 169(3), pp.929~935 (1994)
- A. R. Jamali, "Some new non-abelian 2-groups with abelian automorphism groups", J. Group Theory, 5(1), pp.53~57 (2002)
- L. -C. Kappe, "Groups with 3-abelian normal closures", Arch. Math. (Basel), 51(2), pp.104~110 (1988)
- M. R. R. Moghaddam, "Some properties of 2-auto-Engel groups", ~
- Mohammad Reza R. Moghaddam, "The lower autocentral series of Abelian groups", 대한수학회보, 48(1), pp.79~83 (2011), 대한수학회
- M. R. R. Moghaddam, "Some properties of autocentral automorphisms of a group", Ricerche Mat, 59(2), pp.257~264 (2010)
- D. J. S. Robinson, "Finiteness Conditions and Generalized Soluble Groups, Parts 1 and 2" (1972), Springer-Verlag
- A. Tortora, "Some properties of Bell groups", Comm. Algebra, 37(2), pp.431~438 (2009)