1.3.7 Permutation group

Problem: Given is the goup \( G=(a, b) \) and its subgroup \( H \). 1. Find the order of elements \( a, b, a b \). 2. Determine the order \( H \) and make a Cayley table for it, 3. Check that \( b H=H b \), and deduce from this that the subgroup \( H \) is normal in \( G \). 4. Describe the cosets \( G \) by \( H \). 5. Check that \( G / H \) is cyclic, and find its order. 6. Determine the order of the group \( G \). 7. Determine whether a cyclic subgroup with generator \( b \) is normal in \( G \), 8. Find all subgroups \( Z(b) \). \[ a=\left(\begin{array}{lll} 1 & 2 & 3 \end{array}\right), \quad b=\left(\begin{array}{ll} 12 \end{array}\right)(45), \quad H=(a) \text {. } \]