📂Topological Data Analysis

Free groups in Abstract Algebra

Definition ¹

1. Given an index set $I \ne \emptyset$, let’s refer to the set $A := \left\{ a_{i} : i \in I \right\}$ as the alphabet, and its elements $a_{i} \in A$ as letters.
2. For an integer $n \in \mathbb{Z}$, expressions like $a_{i}^{n}$ are referred to as syllables. A finite juxtaposition of these, $w$, is called a word.
3. A syllable $a_{i}^{n} a_{i}^{m}$ can be represented as $a_{i}^{n+m}$, this is called elementary contraction. A word that can no longer undergo elementary contraction is called a reduced word, especially $1 := a_{i}^{0}$ is called empty word.
4. Let’s denote by $F [A]$, the set of all reduced words that can be formed with letters from the alphabet $A$. Define a binary operation $\cdot : F[A]^{2} \to F[A]$ on two words $w_{1} , w_{2} \in F[A]$ such that $w_{1} \cdot w_{2}$ appears in reduced form. The group $\left( F[A], \cdot \right)$ is called the Free group generated by $A$.
5. If $G$ is a group with elements of set $A := \left\{ a_{i} : i \in I \right\}$ as generators, and there exists an isomorphism $\phi : G \to F [A]$ with $\phi \left( a_{i} \right) = a_{i}$, then we say that $G$ is Free on $A$, and call $a_{i}$ the free generators of $G$.
6. A group that is free on a set $A \ne \emptyset$ is defined as a Free group, and the cardinality $|A|$ of set $A$ is called the rank of the free group.

Explanation

Despite the lengthy definitions, everything becomes straightforward with examples. Don’t be alarmed by terms like ‘alphabet’ or ‘word’. The term algebra itself refers to the study of substituting numbers with letters. Considering this, the approach of applying operations to sets might have seemed too abstract at first. In fact, after defining free groups, the terms mentioned in definition 4 are hardly used. Let’s comfortably look at some examples.

Alphabet and Letters

$$A = \left\{ a, b \right\}$$

Considering the above alphabet, there are only two letters, $a$ and $b$.

Syllables and Words

For the alphabet $A$, $$a^{2} , b^{3}, b^{-1}$$ are all syllables. They are mentioned in the sense of being listed finitely and with duplication allowed, hence the term juxtaposition is used, and by definition, they are simply called words. $$a^{2} b \\ bbab \\ b^{-2} a a a^{-2} b a^{-24}$$

Reduced Words and Empty Word

As an example, let’s examine the process by which the last word $b^{-2} a a a^{-2} b a^{-24}$ is reduced. $$\begin{align*} & b^{-2} a a a^{-2} b a^{-24} \\ =& b^{-2} a^{2} a^{-2} b a^{-24} \\ =& b^{-2} a^{0} b a^{-24} \\ =& b^{-2} 1 b a^{-24} \\ =& b^{-2} b a^{-24} \\ =& b^{-1} a^{-24} \end{align*}$$ Here, $a^{0} = 1$ functions much like an identity element, and indeed, it’s called the Empty Word. After becoming a group, we don’t necessarily have to write $1$ in the alphabet.

Free group generated by $A$

From the buildup so far, $\left( F[A], \cdot \right)$ naturally becomes a group. The identity element is the empty word $1$, and for every word $w$, there exists an inverse $w^{-1}$ that satisfies the following. $$w \cdot w^{-1} = w^{-1}\cdot w = 1$$ Originally, $F[A]$ is a group because specific identity and inverses are given. Thus, a free group is essentially ‘a group made to inevitably be a group’.

groups Free on $A$

$$F[A] = \left\{ \cdots , a^{-2} b^{-1} , a^{-1} b^{-1}, a^{-1}, b^{-1} , 1 , a , b , ab , a b a \cdots \right\}$$ Listing the elements of $F[A]$ as shown above. Although the definitions so far are intuitive and easy to understand, our interest actually lies in the general case of $G$. For instance, considering the group of integers $\left( \mathbb{Z} , + \right)$, since it’s a cyclic group with elements of $\left\{ 1 \right\}$ as generators and isomorphic to $F \left[ \left\{ a \right\} \right]$, it can be said to be free on $\left\{ a \right\}$.

Free groups and Rank

From the examples so far, $\mathbb{Z}$ is free on a singleton set $\left\{ a \right\}$ thus has a rank of $1$, and the free group generated by $A = \left\{ a,b \right\}$ is of rank $2$.