Permutation 2-groups I: Structure and splitness

Permutation 2-groups I: Structure and Splitness

By a 2-group we mean a groupoid equipped with a weakened group structure. It is called split when it is equivalent to the semidirect product of a discrete 2-group and a oneobject 2-group. By a permutation 2-group we mean the 2-group Sym(G) of self-equivalences of a groupoid G and natural isomorphisms between them, with the product given by composition of self-equivalences. These generalize the ...

QUASI-PERMUTATION REPRESENTATIONS OF METACYCLIC 2-GROUPS

By a quasi-permutation matrix we mean a square matrix over the complex field C with non-negative integral trace. Thus, every permutation matrix over C is a quasipermutation matrix. For a given finite group G, let p(G) denote the minimal degree of a faithful permutation representation of G (or of a faithful representation of G by permutation matrices), let q(G) denote the minimal degree of a fa...

Synchronization 2: Permutation groups

Notation Ω will denote a set (often the set The image of an element v ∈ Ω under a permutation g of Ω will be denoted by vg. So if g and h are permutations, and we define composition by the rule that gh means " apply g, then h " , then v(gh) = (vg)h.

I . Permutation Groups and Coherent ( Cellular ) Algebras

quasi-permutation representations of metacyclic 2-groups

by a quasi-permutation matrix we mean a square matrix over the complex field c with non-negative integral trace. thus, every permutation matrix over c is a quasipermutation matrix. for a given finite group g, let p(g) denote the minimal degree of a faithful permutation representation of g (or of a faithful representation of g by permutation matrices), let q(g) denote the minimal degree of a fai...