In the Group $(S_{3},\circ)$, what are the elements of the group $\Big(\big((123)\big), \circ\Big)$?

Question

Given the Group $(S_{3},\circ)$

What are the elements of the group $\Big(\big((123)\big), \circ\Big)$?

Also, why does $\big((123)\big)$ have two brackets around it?

Answer 1

I suggest avoiding the round-bracket notation for generated substructures at all costs, since they are already used for grouping expressions and delineating function inputs, and using it for anything else almost surely will result in unnecessary ambiguity.

That said, in a group $G$ and given $g \in G$, "$\langle g \rangle$" denotes "$\{ g^k : k \in \mathbb{Z} \}$". In an infinite group, it is important to have both positive powers and negative powers (powers of the inverse), otherwise the set 'generated' may not be a subgroup. In a finite group the inverse of every element is some positive power of that element itself because of Lagrange's theorem, or equivalently because the positive powers must repeat cyclically.

In general, "$\langle x,y,...,z \rangle$" denotes the smallest subgroup of $G$ that contains $x,y,...,z$. We can prove (and you should try) that the smallest such subgroup exists because $G$ is such a subgroup and the intersection of all such subgroups is also such a subgroup. Alternatively, it can be defined to be $\{ \prod_{k=1}^n {a_k}^{c_k} : n \in \mathbb{N} \land \forall k \in [1..n]\ ( a_k \in \{x,y,...,z\} \land c_k \in \mathbb{Z} ) \}$. It is not hard to prove that this is indeed the smallest subgroup we are looking for.