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The general concept of weak-direct product is defined on the following way: A Group G is said to be the weak-direct product of its subgroups H and K when (i) the identity element is the only intersection of H and K and (ii) each element of G is the product of one element in H with one element in K. Semi-direct and direct products are special cases of weak-direct products. When H and K are invariant subgroups, the result is a direct product. When only H is an invariant subgroup, the result is a semi-direct product.
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Two integers are coprime if they share no common positive factors (divisors) except 1.
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The rod group can be obtained by considering the operations of the space group formed by the space group operations, setting all translations (pure and compound) to zero. The rod group is isomorphic to the group of the wavevector k = 0.
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