Book contents
- Frontmatter
- Contents
- Introduction
- A speech in honour of John Cannon and Derek Holt
- Finite groups of Lie type and their representations
- Iterated monodromy groups
- Engel elements in groups
- Some classes of finite semigroups with kite-like egg-boxes of D-classes
- Structure of finite groups having few conjugacy class sizes
- Group theory in cryptography
- A survey of recent results in groups and orderings: word problems, embeddings and amalgamations
- A survey on the minimum genus and maximum order problems for bordered Klein surfaces
- On one-relator quotients of the modular group
- Miscellaneous results on supersolvable groups
- Automorphisms of products of finite groups
- A rational property of the irreducible characters of a finite group
- Automotives
- On n-abelian groups and their generalizations
- Computing with matrix groups over infinite fields
- Trends in infinite dimensional linear groups
- Engel conditions on orderable groups and in combinatorial problems (a survey)
- References
Some classes of finite semigroups with kite-like egg-boxes of D-classes
Published online by Cambridge University Press: 05 July 2011
- Frontmatter
- Contents
- Introduction
- A speech in honour of John Cannon and Derek Holt
- Finite groups of Lie type and their representations
- Iterated monodromy groups
- Engel elements in groups
- Some classes of finite semigroups with kite-like egg-boxes of D-classes
- Structure of finite groups having few conjugacy class sizes
- Group theory in cryptography
- A survey of recent results in groups and orderings: word problems, embeddings and amalgamations
- A survey on the minimum genus and maximum order problems for bordered Klein surfaces
- On one-relator quotients of the modular group
- Miscellaneous results on supersolvable groups
- Automorphisms of products of finite groups
- A rational property of the irreducible characters of a finite group
- Automotives
- On n-abelian groups and their generalizations
- Computing with matrix groups over infinite fields
- Trends in infinite dimensional linear groups
- Engel conditions on orderable groups and in combinatorial problems (a survey)
- References
Summary
Abstract
In this article, by considering some presentations of groups and semigroups, we investigate the structure of the groups and semigroups presented by them and introduce some infinite classes of semigroups which have kite-like egg-boxes of D-classes. These semigroups have a unique idempotent and the minimal two-sided ideal of them is isomorphic to the group presented by the same presentation as for these semigroups. All of the Green's relations in these semigroups coincide and every proper subsemigroup of them is a subgroup.
Introduction
Let π be a semigroup and/or group presentation. To avoid confusion we denote the semigroup presented by π by Sg(π) and a group presented by π by Gp(π).
The class of deficiency zero groups presented by
has been studied in [4] where the corresponding group has been proved to be finite of order
for every integer n ≥ 2, where ⌊t⌋ denotes the integer part of a real t and is the sequence of Lucas numbers
In [4], it has been proved that all of these groups are metabelian and that if n ≡ 0 (mod 4) or n ≡ ±1 (mod 6) they are metacyclic.
Also, for every integer n ≥ 2, the presentations
and
of semigroups have been studied in [2] by the authors of this article and in that investigation, their relationship with Gp(πi) has been found as follows:
Theorem 1.1For every n ≥ 2, |Sg(π2)| = |Gp(π2)| + n - 1.
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- Information
- Groups St Andrews 2009 in Bath , pp. 118 - 123Publisher: Cambridge University PressPrint publication year: 2011