Definition of the Group

According to the group theory, the symmetry operations are the members of a group if they satisfy the following group axioms:

1. The successive application of two operations is equivalent to the application of a member of the group. In other words, if the operations $A$ and $B$ belong to the same group then $A\cdot B = C$, where $C$ is also the operation from the same group. Note, that in general $A\cdot B \ne B\cdot A$.
2. One of the operations in the group is the identity operation $E$. This means that $A\cdot E=E\cdot A=A$.
3. The reciprocal of each operation is a member of the group: if $A$ belongs to a group, then $A^{-1}=B$, where $B$ is also the member of the group. Note, that $A\cdot A^{-1}=A^{-1}\cdot A=E$.
4. Multiplication of the operations is associative: $A\cdot B\cdot C = (A\cdot B)\cdot C = A\cdot (B\cdot C)$.

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