What is the generated set $\langle a,b\rangle$?

Question

Can someone please explain what this notation means in group theory? If $\langle a\rangle$ is all the 'multiples' of $a$, then what on earth is $\langle a,b\rangle$?

Answer 1

As ancient mathematician pointed out, $\langle a,b\rangle$ is the smallest subgroup containing the elements $a$ and $b$. This is the definition of the notation.

Consider the set $\{g_1g_2...g_n : n \in \mathbb{N}, g_i \in \{a,b,a^{-1},b^{-1}\}\}$. Let's call this set $S$. Firstly note that this set is exactly all the combinations of products of $a,b$ and their inverses. It is not hard to see that $e \in S$, $S$ is closed with respect to the operation, and $S$ is closed with respect to inverses. This means that $S$ is a subgroup of our underlying group $G$ (and obviosuly $S$ contains $a,b$).

Now consider $H$, which we define to be any subgroup of $G$ that contains $a$ and $b$. Note that any element of $S$ is also an element of $H$ (because subgroups are closed with respect to the operation and inverses). So this means that $S \subset H$.

But this means that $S$ is a subset of any subgroup containing both $a$ and $b$, and $S$ is also a subgroup containing $a,b$! In other words, $S$ is the smallest subgroup of $G$ containing both $a,b$. This means we have proven that the set $\langle a,b \rangle$ is actually the set $S$.

The point here is that the notation $\langle a,b\rangle$ was defined to be the smallest subgroup containing $a$ and $b$, and then we proved that such a subgroup exists, and in fact is equal to the set $S$. So now whenever we see $\langle a,b \rangle$, we can just replace it with the set $S$.