Prove that for a group the inverse of each group element and the identity are unique. Prove that the inverse of the product of two group elements is given by \((a \cdot b)^{-1}=\) \(b^{-1} a^{-1}\). Check this against the product \(a \cdot b=(12) \cdot(23)\) from Table 2.2 .

Data from Table  2.2

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Table 2.2. Multiplication table A - B for S+ A\B e (12) (23) (13) (123) (321) e e (12) (23) (13) (123) (321) (12) (12) e (321) (123) (13) (23) (23) (23) (123) e (321) (12) (13) (13) (13) (321) (123) e (23) (12) (123) (123) (23) (13) (12) (321) e (321) (321) (13) (12) (23) e (123) *Entries A B, where the left column gives A and the top row gives B. +