How many generators do a 4-strands braid group have?

Question

I know that this might be a trivial question, but, I'm asking that because source on the topic tend to give different presentations.
Maybe it's just me not noticing the difference, but I'd like to point it out bacuase I think it can be helpful to avoid confusion for someone who is approaching braid groups. The problem lies in what is considered generators of abraid group, in particular a 4-strands to be specific. Are inverses of the 3 "standard" braids included in the set of generators?($\displaystyle \sigma 1$, $\displaystyle \sigma 2$, $\displaystyle \sigma 3$).

For example Wikipedia article talks about 3 generators while a bunch of other sources include the inverses ($\displaystyle \sigma 1^{-1}$, $\displaystyle \sigma 2^{-1}$, $\displaystyle \sigma 3^{-1}$) in the set of generators. My intuition tells me to include them too, but I'm afraid I'm missing ssomething.

Answer 1

Definition. Let $S\subset G$. Then the subgroup generated by $S$, written $\langle S\rangle$, is the smallest subgroup of $G$ containing $S$. Formally: $\langle S\rangle:=\displaystyle\bigcap_{S\subset H,\\ H\leq G}H$.

Then a generating set of $G$ is a subset $S$ of $G$ where $\langle S\rangle=G$. For example, $\langle 1\rangle=\mathbb{Z}$.

There is nothing in this definition about "products of elements". However, it turns out that the following holds:

A set $S$ generates $G$ if and only if every element $g\in G$ can be written as a product of elements from $S\cup S^{-1}$ (where $S^{-1}:=\{s^{-1}\mid s\in S\}$ is the set of inverses of elements of $S$).

The phrase from Wikipedia "The group $\langle S\rangle$ may be viewed as the product of all elements of $S$ and their inverses" is implying that the inverses of $S$ are not necessarily in $S$! For example, every integer can be written as a sum of $1$s and of $-1$s, so $\langle 1\rangle=\mathbb{Z}$. To be clear: there is no assumption that $S=S^{-1}$.

Lets end on an exercise, which basically says you can ignore this subtlety for finite groups:

Exercise. Prove that set $S$ generates $G$ if and only if every element $g\in G$ can be written as a product of elements from $S$.