Book contents
- Frontmatter
- Contents
- Introduction
- Galois groups through invariant relations
- Construction of Co3. An example of the use of an integrated system for computational group theory
- Embedding some recursively presented groups
- The Dedekind-Frobenius group determinant: new life in an old problem
- Group characters and π-sharpness
- Permutation group algorithms via black box recognition algorithms
- Nonabelian tensor products of groups: the commutator connection
- Simple subalgebras of generalized Witt algebras of characteristic zero
- Applications of the Baker-Hausdorff formula in the theory of finite p-groups
- Generalizations of the restricted Burnside problem for groups with automorphisms
- The ∑m-conjecture for a class of metabelian groups
- Rings with periodic groups of units II
- Some free-by-cyclic groups
- The residually weakly primitive geometries of the Suzuki simple group Sz(8)
- Semigroup identities and Engel groups
- Groups whose elements have given orders
- The Burnside groups and small cancellation theory
- Solvable Engel groups with nilpotent normal closures
- Nilpotent injectors in finite groups
- Some groups with right Engel elements
- The growth of finite subgroups in p-groups
- Symplectic amalgams and extremal subgroups
- Primitive prime divisor elements in finite classical groups
- On the classification of generalized Hamiltonian groups
- Permutability properties of subgroups
- When Schreier transversals grow wild
- Probabilistic group theory
- Combinatorial methods: from groups to polynomial algebras
- Formal languages and the word problem for groups
- Periodic cohomology and free and proper actions on ℝn × Sm
- On modules over group rings of soluble groups of finite rank
- On some series of normal subgroups of the Gupta-Sidki 3-group
Some groups with right Engel elements
Published online by Cambridge University Press: 04 August 2010
- Frontmatter
- Contents
- Introduction
- Galois groups through invariant relations
- Construction of Co3. An example of the use of an integrated system for computational group theory
- Embedding some recursively presented groups
- The Dedekind-Frobenius group determinant: new life in an old problem
- Group characters and π-sharpness
- Permutation group algorithms via black box recognition algorithms
- Nonabelian tensor products of groups: the commutator connection
- Simple subalgebras of generalized Witt algebras of characteristic zero
- Applications of the Baker-Hausdorff formula in the theory of finite p-groups
- Generalizations of the restricted Burnside problem for groups with automorphisms
- The ∑m-conjecture for a class of metabelian groups
- Rings with periodic groups of units II
- Some free-by-cyclic groups
- The residually weakly primitive geometries of the Suzuki simple group Sz(8)
- Semigroup identities and Engel groups
- Groups whose elements have given orders
- The Burnside groups and small cancellation theory
- Solvable Engel groups with nilpotent normal closures
- Nilpotent injectors in finite groups
- Some groups with right Engel elements
- The growth of finite subgroups in p-groups
- Symplectic amalgams and extremal subgroups
- Primitive prime divisor elements in finite classical groups
- On the classification of generalized Hamiltonian groups
- Permutability properties of subgroups
- When Schreier transversals grow wild
- Probabilistic group theory
- Combinatorial methods: from groups to polynomial algebras
- Formal languages and the word problem for groups
- Periodic cohomology and free and proper actions on ℝn × Sm
- On modules over group rings of soluble groups of finite rank
- On some series of normal subgroups of the Gupta-Sidki 3-group
Summary
Abstract
In 1970 I.D. Macdonald exhibited a nilpotent group in which the square and the inverse of a right 3-Engel element need not be 3-Engel and thereby showing that the set of right 3-Engel elements of a group need not form a subgroup. In this note a nilpotent group for each n ≥ 3 is constructed such that the set of right n-Engel elements in each group is not a subgroup.
Introduction
An element a of a group is called a right n-Engel element, n a positive integer, if for each element g of the group [a, ng] = 1 (cf. [Rob72, p. 40]). Commutators are written left-normed and repeated entries in a commutator are indicated by left subscripts. Clearly, the set of right 1-Engel elements is the centre of the group and therefore a subgroup. W. Kappe [Kap61] proved that the set of right 2-Engel elements of a group is also a subgroup. I.D. Macdonald [Mac70] showed that the set of right 3-Engel elements of a group need not form a subgroup by constructing a group with an element that is right 3-Engel but whose inverse and square are not. In this paper we will construct a group for each n ≤ 3 with a right n-Engel element whose inverse and square are not right n-Engel. This answers a question raised at the conference by W. Kappe for such an example for n = 4.
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- Groups St Andrews 1997 in Bath , pp. 571 - 578Publisher: Cambridge University PressPrint publication year: 1999
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