Condensed matter physics MOC

Crystallographic group

A crystallographic group or space group in dimension is a discrete subgroup of the Euclidean group containing a normal -lattice subgroup such that quotient group , called the Point group, is a subgroup of the real orthogonal group . #m/def/group This gives the following pair of short exact sequences

https://q.uiver.app/#q=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

Such a group is the symmetry group of a Crystal. According to the Crystallographic restriction theorem, point groups must have rotational symmetries that are 1-,2-,3-,4-, or 6-fold.

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