Primitive pairs of K-groups

TL;DR

This paper extends previous work on primitive pairs of groups by weakening the $p$-solvable condition to a $K$-group hypothesis, leading to new proofs of the Signalizer Functor Theorem.

Contribution

It introduces a broader $K$-group framework for primitive pairs, enabling new proofs of fundamental theorems in group theory.

Findings

Proves a non-existence theorem for certain amalgams of $K$-groups.

Provides a new proof of the Nonsolvable Signalizer Functor Theorem.

Extends previous results from $p$-solvable groups to $K$-groups.

Abstract

In 'Primitive pairs of $p$-solvable groups', J. Algebra 324 (2010) 841-859, the author proved a non existence theorem for certain types of amalgams of $p$-solvable groups in the presence of operator groups acting coprimely on the groups in the amalgam. An application of that work was a new proof of the Solvable Signalizer Functor Theorem. In this article, the $p$-solvable restriction will be weakened to a $K$-group hypothesis. An application of this work will be a new proof of the Nonsolvable Signalizer Functor Theorem.