Embedding properties in direct products

Summary

Introduction

This paper is a survey article containing an up-to-date account of recent achievements regarding embedding properties in direct products of groups. In the last years, several authors are carrying out a systematic study with the aim of understanding how subgroups with various embedding properties can be detected and characterized in the subgroup lattice of a direct product of two groups in terms of the subgroup lattices of the two groups.

Unless otherwise stated all groups considered in this paper are finite.

Direct products are maybe the easiest way to construct new groups from given ones and in spite of the simplicity of this construction, their structures are sometimes surprising.

The subgroup structure of direct products is well-known by a classical result due to Goursat. In this paper G₁ × G₂ = {(g₁, g₂) | g_i ∈ G_i, i = 1, 2} will always denote the direct product of the groups G₁ and G₂ and π_i will denote the canonical projection πi : G₁ × G₂ → Gi, for i = 1, 2. For a subgroup U of G₁ × G₂:

• π_i (U) = UG_j ∩ Gi, {i, j} = {1, 2},

• U ∩ G_i ⊴ π_i (U), for i = 1, 2.

Goursat's theorem states that, apart from the direct product of subgroups of the direct factors, only ‘diagonal’ subgroups appear in a direct product.