Automorphism groups of superextensions of groups

TL;DR

This paper explores the structure of superextensions of groups, proving that group isomorphism is equivalent to superextension isomorphism and describing automorphism groups for small groups.

Contribution

It establishes a fundamental link between group isomorphisms and their superextensions, and characterizes automorphism groups for small groups.

Findings

Group isomorphism iff superextension isomorphism

Automorphism groups of superextensions for groups of size ≤ 5

Extension of binary operations to superextensions

Abstract

The superextension $\lambda(X)$ of a set $X$ consists of all maximal linked families on $X$. Any associative binary operation $*: X\times X \to X$ can be extended to an associative binary operation $*: \lambda(X)\times\lambda(X)\to\lambda(X)$. In the paper we study isomorphisms of superextensions of groups and prove that two groups are isomorphic if and only if their superextensions are isomorphic. Also we describe the automorphism groups of superextensions of all groups of cardinality $\leq 5$.