How to Build Groups

Introduction

Most works on finite groups consider given but unknown groups, and analyse them by dismantling them, perhaps starting by considering their Sylow subgroups. This page takes the opposite approach. It discusses how to build groups up from smaller ones.

Suppose we are interested in groups of order 18, and want to list them all. The usual approach is to say "suppose we have a group of order 18. Its Sylow 3-subgroup can be either C9 or C3×C3 ... ", and to use a variety of methods to pull it apart. The approach taken here is to consider all the groups of orders 2, 3, 6 and 9 (it is assumed that we are already acquainted with these), and to examine all the ways of putting them together to build a group of order 2×9 or 3×6.

This approach has two weaknesses. One is that it will never be able to build a simple group. The other is that (to the best of my knowledge) the number of ways of putting two known groups together to build a bigger one is infinite; however, a few of them will get us a long way.

Section 0: Cyclic groups

Before we can start putting groups together, we must have some groups. We start with the cyclic groups: C1, C2, C3, C4, etc. We will use these as the building blocks for all other groups. I assume that the reader understands cyclic groups.

All cyclic groups are Abelian.

We now have 31 of the 93 groups of order less than 32.

Section 1: Direct Products.

Given two groups P and Q, we can build their direct product P×Q as follows. If P is {p₁, p₂, .. p_n; +} and Q is {q₁, q₂, .. q_m; *}, then the elements of their direct product, denoted by P×Q; are the ordered pairs (p_i, q_j) and its operation is such that (p_i, q_j)(p_k, q_l) = (p_i+p_k, q_j*q_l). P×Q has both P and Q as normal subgroups.

The direct product of two Abelian groups is Abelian. Therefore all the groups we can build so far are Abelian. We can in fact now build all finite Abelian groups. We can build 48 of the 93 groups of order less than 32. However they are least interesting ones.

Section 2: Semidirect Products.

See sub-page on building semidirect products.

Section 2, Summary

Now that we can build direct and semidirect products, we can starting from cyclic groups build 88 of the 93 groups of order less than 32. The five that we cannot build are Q8, Q8×C2, Q16, Q8×C3, and Q8⋊C3. We need a way to build (among other things) quaternion groups.

Interlude: Groups of order 15.

At around this point, works on finite groups traditionally consider all the groups of order 15.

The prime factors of 15 are 3 and 5, so any group of order 15 is built from groups of orders 3 and 5. These component groups are unique: C3 and C5. Their automorphism groups are C2 and C4 respectively. The only homomorphisms from C5 to C2 and from C3 to C4 are the trivial ones that lead to C3×C5 (isomorphic to C15), so this is the only group of order 15 that we are able to build. Conceivably, there is another way of building a group from two smaller ones, which allows us to build a different group of order 15. In fact there isn't.

See sub-page on building toll-bean extensions.

Section 3, Summary

Now we have three ways of putting groups together:

• direct products, denoted by ×
• semi-direct products, denoted by ⋊
• toll-bean extensions. We shall denote these by ↑

Hypothesis: using these three methods, and starting from prime cyclic groups, we can build all finite solvable groups.

Stronger hypothesis: using these three methods, and starting from simple groups, we can build all finite groups

More pages on groups