# What is the difference between space and point groups?

kompabt
According to wikipedia:
"A crystallographic point group is a set of symmetry operations, like rotations or reflections, that leave a point fixed while moving each atom of the crystal to the position of an atom of the same kind."
"The space groups in three dimensions are made from combinations of the 32 crystallographic point groups with the 14 Bravais lattices."

I don't understand the second part of the last sentence. (WITH the 14 Bravais lattices) And if I combine point groups why won't I also get other point groups?

## Answers and Replies

Science Advisor
Space groups are all groups of transformations which map an (infinite ideal) crystal onto itself. Most of the elements of this group also shift the crystal (that's the translational part). Now one can consider classes of symmetry elements whose effect is equivalent up to some translation. The symmetry groups spanned by these classes are the point groups.

Useful nucleus
In mathematical terms, every element of the space group can be written as (t$$\phi$$) , where t is a translation and $$\phi$$ is an orthogonal linear operator. Then, the corresponding element of the point group would be $$\phi$$.
So If G is a space group, define the function "remove the translation part", the image of that function is called the point group.

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