Notation question: Subroup generated by two elements?

Question

Let there be $H$ subgroup of symmetric group $S_4$, so that $H= \langle (12)(34),(234) \rangle$. What does the notation $\langle (12)(34),(234) \rangle$ mean? I know that if there's one elements, then it's all the powers of that element.

Thanks in advance!

Answer 1

This is the smallest subgroup that contains both $(12)(34)$ and $(234)$.

This object is well defined as the intersection of all subgroups that contains $(12)(34)$ and $(234)$ because intersection of subgroups is a subgroup.

Another description:

$$ H= \{ a_1a_2\cdots a_n: n\in\Bbb N, \forall k\ \ a_k\in\{ (12)(34),(234) \} \} $$