# Permutations & transformations of finite and infinite sets

**Date**: Wednesday 31 May 2017, 16:00 – 17:00**Location**: Mathematics Level 8, MALL 1, School of Mathematics**Type**: Pure Mathematics seminars, Algebra, logic and algorithms seminars, Logic seminars, Seminar series**Cost**: Free

#### Yann Peresse, University of Hertfordshire. Part of the algebra, logic and algorithms seminar series.

The permutations of the elements of a set X, i.e. the bijective functions from X to X, form a group under composition of functions, called the 'symmetric group' or Sym(X). Symmetric groups play a fundamental and prominent role in both Group Theory and Combinatorics. The equivalent object in Semigroup Theory is the 'full semigroup transformation semigroup' Trans(X) of all transformation of X, i.e. of all functions from X to X. In addition to Semigroup Theory, the full transformation semigroup also appears in theoretical Computer Science, for example in connection with the Czerny conjecture.

In this talk, we will explore some fundamental algebraic and combinatorial properties of Sym(X) and Trans(X), such as the least size of a generating set or the structure of their sub(semi)group lattice. We will compare and contrast the cases when X is finite and when X is infinite. As such, we will sometimes obtain the "usual" combinatorial type results but sometimes answers involving topology or uncountable cardinals. Despite some of these slightly unusual looking answers, the talk will be approachable and is aimed at a general 'Mathematics and related areas' audience.

*All are very welcome. Tea/coffee in the Staff Common Room beforehand at 3.45pm.
Organised by Olaf Beyersdorff and Vincenzo Mantova.*

**Yann Peresse,** *University of Hertfordshire*