1.6.30 Fields, Groups, Rings

Problem: Let \( H \) be a subgroup, generated by the element \( b \) in the multiplicative \( \mathbb{Z}_{p}^{*} \) of residues modulo \( p \), and the \( g H \) coset of the subgroup \( H \), generates by the element \( g \). Calculate the subgroup \( H \) and the coset \( g H \). \[ p=97, b=8, g=2 \text {. } \]