Abstract: If X is an object such that the notion of an automorphism of X is defined (e.g.,
an algebraic structure, a graph, a topological space, etc.), then one can define an
equivalence relation ∼ on X via x ∼ y if and only if α(x) = y for some automorphism
α of X. The equivalence classes of ∼ are called the automorphism orbits of X.
Say that X is highly symmetric if and only if all elements of X lie in the same
automorphism orbit. Finite highly symmetric objects are studied across various
mathematical disciplines, e.g. in combinatorics, graph theory and geometry. When
X is taken to be a finite group G, this problem is not really interesting, as G is highly
symmetric in the above sense if and only if G is trivial (i.e., G has only one element).
However, the following weaker property is interesting to study: For each ρ ∈ (0, 1],
say that G is ρ-symmetric if and only if G admits an automorphism orbit of size at
least ρ|G|. In this talk, we discuss results, due to the author, that give structural
restrictions (in terms of ρ) on ρ-symmetric finite groups G.