I have searched days for the answer of this question, but without success. Some background information:
A point group must fulfil the mathematical requirements of a group:
- The product of two operations is another operation in the group
- The identity element (I) is present
- The existence of an inverse operator $RR^{-1} = 1$
- Associative multiplication of operations*
The definition of the reciprocal lattice is: All vectors $\vec{G}$ for which $e^{i\vec{G}*\vec{T}}=1$ for all translation vectors $\vec{T}$ of the translation lattice.
Let P be a point-group operation of the direct lattice (To stress, the original lattice $\vec{R}=a_1\vec{n_1}+a_2\vec{n_2}+a_3\vec{n_3},$ from which we start), i.e., $\vec{T'}=P\vec{T}$ are elements of the direct lattice for all elements $\vec{T}$ of the direct lattice. The inner product has the following property: $\vec{G}*({P\vec{T}})=(P^{-1}\vec{G})*{\vec{T}}$.
The question: Argue now that the point group of the reciprocal lattice is the same as the point group of the direct lattice. In other words, show that if P is a point-group operation of the direct lattice, then P is also a point-group operation of the reciprocal lattice.
Hint: For all $g$ in the group there is an element $g^{-1}$, the inverse of $g$, such that $g•g^{-1}=g^{-1}•g=e$.
I hope I have given sufficient information to answer the question. Any help would be much appreciated.