Generating set - inconsistency?

Question

In my lecture notes $‹S›$ is defined as follows:

Then later there is this:

But surely this is exactly what $‹s,t›$ is? Directly from the Proposition, with $S=${$s,t$}, $H=${$s^jt^k$} with $s,t∈S$ and $j,k∈Z$ and then $H=‹S›=‹s,t›$.

Surely then these two sections contradict each other? Or more likely, I'm missing something here. I'm guessing it's something like ‹s,t› is different from ‹{s,t}› since the former is a pair of elements and the latter is a set. But my lecture notes only define it as a set, never as just an element or a pair of elements. Can anyone shed any light here?

Answer 1

In the proposition, it's possible that $s_i = s_j$ with $i\neq j$. So, for example, it's possible that $sts \neq s^jt^k$ for any $j$ or $k$. Then $sts$ is in $\left<s,t\right>$ but not $\left\{ s^jt^k\,:\,j,t\in \mathbb{Z}\right\}$

For a concrete example, just consider the free group on two generators.

And to clarify for your guess, $\left<s,t\right>$ is the same as $\left<\left\{s,t\right\}\right>$