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I have been reading D.R. Farkas’ “Crystallographic groups and their mathematics“, which seems like a reference introduction to the subject.

In it, point groups are defined as the quotient of the space group by its subgroup of translations. This subgroup is finite, and hence it fixes a point, defined for example as the average over the orbit of any point.

This seems to me to imply that all space group elements are the product of an element of the point group and lattice translations, hence that all space groups are split/symmorphic, which is not the case.

Another source 1 claims that the point group is not a subgroup of the orthogonal group in general, and yet claims that it is always finite, (so the construction of the fixed point by the average should work), which is where the contradiction seems most evident to me.

I am certainly missing a detail to resolve this contradiction. Any help would be greatly appreciated!

user288227
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    You wrote ""this subgroup is finite " but it's not a subgroup, it's a quotient group. – Derek Holt Jan 22 '20 at 15:29
  • Ah yes! Thanks a lot! Add this as an answer and I’ll validate it. – user288227 Jan 22 '20 at 15:57
  • If both the glide and the reflection of a glide reflection are symmetries, then the glide reflection mod translations is just a reflection, but if the glide and reflection part of a glide reflection are not themselves symmetries then the glide reflection is not equivalent to a reflection mod translations. This might be relevant. – anon Nov 04 '20 at 22:37

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