2
$\begingroup$

In mathematics, a group $G$ is called free if there is a subset $S$ of $G$ such that any element of $G$ can be written in one and only one way as a product of finitely many elements of $S$ and their inverses.

The group $(\mathbb{Z},+)$ of integers is free; we can take $S = \{1\}$. ("Free group", Wikipedia)

By saying "product of finitely many elements of $S$ and their inverses", in case of $(\mathbb{Z},+)$, does product refer to $+$?

If not, can anyone correct misunderstanding?

  • 4
    Yes product refers to the law of the group, even when it's the usual addition.2012-05-31
  • 1
    Yes; the product just refers to the group operation. For abelian groups, $+$ is often used instead of $\cdot$ or mere juxtaposition.2012-05-31
  • 1
    Note: When we talk about a "free group on X", then $X$ is a set whose elements correspond to the $S$ of the definition. So a "free group on $\mathbb{Z}$" would be a free group of countable rank, rather than the free group $\mathbb{Z}$.2012-05-31

2 Answers 2