The half-automorphism group of code loops

TL;DR

This paper characterizes the structure of half-automorphism groups in code loops, showing their relation to automorphism groups and identifying limitations on elementary mappings based on loop rank.

Contribution

It provides a detailed description of the half-automorphism group structure of code loops and establishes rank-based restrictions on elementary half-automorphisms.

Findings

Half-automorphism group is the product of automorphism group and an elementary abelian 2-group.

Elementary mappings are only half-automorphisms for code loops of rank ≤ 3.

The structure of half-automorphism groups is explicitly characterized.

Abstract

For any code loop $L$, we prove that the half-automorphism group of $L$ is the product of the automorphism group of $L$ by an elementary abelian $2-$group consisting of all half-automorphisms that acts as the identity on a fixed basis. Also, we prove that elementary mappings only can be a half-automorphism on code loops of rank at most $3$.