Small loops of nilpotency class three with commutative inner mapping groups

TL;DR

This paper constructs small loops of Cs"org"H{o} type with nilpotency class three and non-elementary abelian inner mapping groups, expanding the understanding of loops with commuting inner mappings beyond class two.

Contribution

It provides explicit constructions of small loops of Cs"org"H{o} type with nilpotency class three and non-elementary abelian inner mapping groups, using a detailed group-theoretic setup.

Findings

Loops of Cs"org"H{o} type can have order at least 128.

Such loops have inner mapping groups that are not elementary abelian.

The paper describes all nontrivial setups with order 128.

Abstract

Groups with commuting inner mappings are of nilpotency class at most two, but there exist loops with commuting inner mappings and of nilpotency class higher than two, called loops of Cs\"org\H{o} type. In order to obtain small loops of Cs\"org\H{o} type, we expand our programme from `Explicit constructions of loops with commuting inner mappings', European J. Combin. 29 (2008), 1662-1681, and analyze the following setup in groups: Let $G$ be a group, $Z\le Z(G)$, and suppose that $\delta:G/Z\times G/Z\to Z$ satisfies $\delta(x,x)=1$, $\delta(x,y)=\delta(y,x)^{-1}$, $z^{yx}\delta([z,y],x) = z^{xy}\delta([z,x],y)$ for every $x$, $y$, $z\in G$, and $\delta(xy,z) = \delta(x,z)\delta(y,z)$ whenever $\{x,y,z\}\cap G'$ is not empty. Then there is $\mu:G/Z\times G/Z\to Z$ with $\delta(x,y) = \mu(x,y)\mu(y,x)^{-1}$ such that the multiplication $x*y=xy\mu(x,y)$ defines a loop with commuting inner mappings, and this loop is of Cs\"org\H{o} type (of nilpotency class three) if and only if $g(x,y,z) = \delta([x,y],z)\delta([y,z],x)\delta([z,x],y)$ is nontrivial. Moreover, $G$ has nilpotency class at most three, and if $g$ is nontrivial then $|G|\ge 128$, $|G|$ is even, and $g$ induces a trilinear alternating form. We describe all nontrivial setups $(G,Z,\delta)$ with $|G|=128$. This allows us to construct for the first time a loop of Cs\"org\H{o} type with an inner mapping group that is not elementary abelian.