Solvability criteria for finite loops and groups

Summary

Abstract

We study how the solvability of finite loops follows from certain properties of their multiplication groups and inner mapping groups. In this survey we review the progress of this research problem from the last two decades and the best results achieved so far. We also introduce a few new improvements with proofs in order to give the readers some idea about the basic methods used in this study.

Introduction

Let Q be a groupoid with a neutral element e. If the equations ax = b and ya = b have unique solutions x and y in Q for every a, b ∈ Q, then we say that Q is a loop. If a loop Q is associative, then it is in fact a group. For each a ∈ Q we have two permutations L_a (left translation) and R_a (right translation) on Q, defined by L_a(x) = ax and R_a(x) = xa for every x ∈ Q. These permutations generate a permutation group M(Q), which is called the multiplication group of Q. Clearly, M(Q) is a transitive permutation group on Q. The stabilizer of the neutral element e is called the inner mapping group of Q and denoted by I(Q). If Q is a group, then I(Q) is just the group of inner automorphisms of Q. The concepts of the multiplication group and the inner mapping group of a loop were defined by Bruck [1] in 1946.

A subloop H of Q is normal in Q if x(yH) = (xy)H, (xH)y = x(Hy) and xH = Hx for every x, y ∈ Q. As in groups, a subloop H of a loop Q is normal if and only if H is the kernel of some homomorphism of Q. A loop Q is said to be solvable if it has a series 1 = Q₀ ⊆ · · · ⊆ Q_n = Q, where Q_i−1 is normal in Q_i and Q_i/Q_i−1 is an abelian group for each i.

Connected transversals

Let G be a group, H ≤ G and let A and B be two left transversals to H in G. We say that the two transversals A and B are H-connected if a⁻¹ b⁻¹ ab ∈ H for every a ∈ A and b ∈ B.