## Mathematics## Seminar |

## Rocky Mountain Algebraic Combinatorics Seminar

### Upcoming Seminars Schedule

### Invariable generation of finite classical groups

Eilidh McKemmie

University of Southern California

We say a group is invariably generated by a subset if it forms a generating
set even if an adversary is allowed to replace any elements with their
conjugates. Eberhard, Ford and Green built upon the work of many others and
showed that, as *n* → ∞, the probability that *S*_{n} is
invariably generated by a random set of elements is bounded away from zero
if there are four random elements, but goes to zero if we pick three random
elements. This result gives rise to a nice Monte Carlo algorithm for
computing Galois groups of polynomials. We will extend this result for *S*_{n}
to the finite classical groups using the correspondence between classes of
maximal tori of classical groups and conjugacy classes of their Weyl groups.

### Covering numbers of finite groups

Michael Epstein

CSU

Any finite noncyclic group is a union of finitely many proper subgroups.
Given such a group G, we define the *covering number* of G to be
the least positive integer m such that G is the union of m proper
subgroups. In general, the problem of determining the covering number of a
given group can be quite difficult; even the covering numbers of
well-studied groups such as the alternating and symmetric groups have not
been completely determined. We will give an overview of some known results
on covering numbers, discuss the connection with pairwise generating sets,
introduce some of the basic methods for finding covering numbers, and if
time permits, I will discuss some of my own recent work on this topic.

Weber 223

4–6 pm

Friday, Nov 1, 2019

(Refreshments in Weber 117, 3:30–4 pm)

Colorado State University

This is a joint Denver U / UC Boulder / UC Denver / U of Wyoming / CSU seminar that meets biweekly. Anyone interested is welcome to join us at a local restaurant for dinner after the talks.