Free Groups, An Introduction

Introduction

A free group is intuitive, but the proofs are somewhat technical. I'll present some intuition first, then get down to business.

A free group on three letters consists of strings of these letters and their inverses. If ABC are the three letters, let (lower case) abc be the inverses. Multiplication in the group is now concatenation, followed by cancellation. Thus AAC * caB becomes AB.

This certainly seems associative. Parentheses shouldn't matter as you tack three words together. Also, every word has an inverse, by reversing and inverting the letters. So it looks like a group.

Technical Definition

A free group is based on a set of generators, or symbols, or letters - different names for the same thing. Each letter is paired with a modified version of itself, which acts as its inverse. Sometimes the letter is written with a bar over it - and sometimes we use upper and lower case, as shown above. The letters and their inverses form an alphabet that will be used to build words of finite length.

A word is reduced if a letter never sits next to its inverse. The reduced words, including the empty word, are the elements of the group.

Group multiplication consists of concatenation followed by cancellation. But is this well defined? We need to show that an unreduced word always leads to the same reduced word, no matter how you proceed.

Suppose an unreduced word yields two different reduced words. Start at one end, one of the reduced words, then proceed step by step through the unreduced word, and wind up at the other reduced word. Each step creates a pair of inverse letters, or cancels a pair of inverse letters. Call these words w₁ through w_n, where w₁ and w_n are distinct reduced words.

Since w₁ and w_n are different words, there are at least two words in the sequence. In fact there are more than two, because w₁ is reduced, and the step from w₁ to w₂ can only create a symbol inverse pair, which means w₂ is not reduced. So there are at least three words in the sequence.

Let the size of a sequence be the sum of the lengths of the words, and assume we have a valid sequence of minimal size. Let w_i be the longest word in the sequence. Moving into w_i creates a pair, and moving on to w_i+1 cancels a pair. If these pairs are independent, reverse the order - cancel first, then create. Now w_i is a shorter word by 4, which is a contradiction.

If the same pair is created and destroyed, remove w_i and w_i+1 to find a smaller sequence. If the two pairs overlap in one letter, the neighboring words are still identical, and you can remove w_i and w_i+1. These are all the cases, hence there is no sequence connecting distinct reduced words. Each unreduced word colllapses to a unique reduced word.

To show associativity, consider (w₁w₂)w₃ and w₁(w₂w₃). In each case the reduced word comes from the unreduced word w₁w₂w₃, hence the parentheses can't change the outcome.

The empty word is the identity element, and the inverse of a word is the reverse of its inverse letters. A group is born.

One Generator

If a free group is generated by X, each word is a sequence of n copies of X, or x. Map these words to the integers n and -n respectively. Verify that this group is isomorphic to Z.