Book contents
- Frontmatter
- Contents
- Introduction
- Gracefulness, group sequencings and graph factorizations
- Orbits in finite group actions
- Groups with finitely generated integral homologies
- Invariants of discrete groups, Lie algebras and pro-p groups
- Groups with all non-subnormal subgroups of finite rank
- On some infinite dimensional linear groups
- Groups and semisymmetric graphs
- On the covers of finite groups
- Groupland
- On maximal nilpotent π-subgroups
- Characters of p-groups and Sylow p-subgroups
- On the relation between group theory and loop theory
- Groups and lattices
- Finite generalized tetrahedron groups with a cubic relator
- Character degrees of the Sylow p-subgroups of classical groups
- Character correspondences and perfect isometries
- The characters of finite projective symplectic group PSp(4, q)
- Exponent of finite groups with automorphisms
- Classifying irreducible representations in characteristic zero
- Lie methods in group theory
- Chevalley groups of type G2 as automorphism groups of loops
On the relation between group theory and loop theory
Published online by Cambridge University Press: 15 December 2009
- Frontmatter
- Contents
- Introduction
- Gracefulness, group sequencings and graph factorizations
- Orbits in finite group actions
- Groups with finitely generated integral homologies
- Invariants of discrete groups, Lie algebras and pro-p groups
- Groups with all non-subnormal subgroups of finite rank
- On some infinite dimensional linear groups
- Groups and semisymmetric graphs
- On the covers of finite groups
- Groupland
- On maximal nilpotent π-subgroups
- Characters of p-groups and Sylow p-subgroups
- On the relation between group theory and loop theory
- Groups and lattices
- Finite generalized tetrahedron groups with a cubic relator
- Character degrees of the Sylow p-subgroups of classical groups
- Character correspondences and perfect isometries
- The characters of finite projective symplectic group PSp(4, q)
- Exponent of finite groups with automorphisms
- Classifying irreducible representations in characteristic zero
- Lie methods in group theory
- Chevalley groups of type G2 as automorphism groups of loops
Summary
Abstract
Loops are nonassociative algebras (and are sometimes known as ‘nonassociative groups’) which can be investigated by using their multiplication groups. This connection to group theory is a source of several interesting group theoretical investigations and in the following four sections we try to cover some of the major problems in this area.
Introduction
Let Q be a groupoid with a neutral element e. If for any a, b ∈ Q each of the equations ax = b and ya = b has a unique solution, then we say that Q is a loop. For each a ∈ Q we have two permutations La (left translation) and Ra (right translation) on Q defined by La(x) = ax and Ra(x) = xa for every x ∈ Q. The permutation group M(Q) generated by the set of all left and right translations is called the multiplication group of Q. It is easy to see that M(Q) is transitive on Q and the stabilizers of elements of Q are conjugated in M(Q). The stabilizer of e ∈ Q is denoted by I(Q) and this stabilizer is called the inner mapping group of Q (it is interesting to observe that if Q is a group, then I(Q) is just the group of inner automorphisms of Q). The concepts of the multiplication group and the inner mapping group of a loop were defined by Bruck [3] in 1946 in an article where he laid the foundation of loop theory.
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- Chapter
- Information
- Groups St Andrews 2001 in Oxford , pp. 422 - 427Publisher: Cambridge University PressPrint publication year: 2003