Find all subgroups of the permutation group \(S_3\)

1. Let \(G\) be a finite commutative group. Show that \[\prod_{g\in G}g^2=e_G.\]

2. Let \(G\) be a group with \(g^2=e_G\) for all \(g\in G.\) Show that \(G\) is commutative.

Let \(H_1, H_2\) be two subgroups of a group \(G.\) Show the union \(H_1 \cup H_2\) is a subgroup of \(G\) if and only if \(H_1\subset H_2\) or \(H_2\subset H_1.\)

Show that a non-empty subset \(H\) of a group \(G\) is itself a group when equipped with the group operation \(*_H=*_G\) and identity element \(e_H=e_G,\) provided for all elements \(a,b \in H,\) we have \(a*_Gb^{-1} \in H.\)