# Are these two space groups in a group–subgroup relationship?

Given two crystal space groups, how can one determine if they are in a group-subgroup relationship? The specific case at hand is P 21/m 2/m 2/a (aka P m m a, #51) and P m m 2 (#25), but knowing how to solve this question in a general manner will sure prove helpful in the future as well…

• I personally don't know too much about crystals or group theory, but it occurs to me that this question is already clear on chemistry and needs help in the math. Migrate to math.SE? (of course, the groups may be only used by chemists, in which case it may be a better fit here) – ManishEarth May 20 '12 at 18:31
• @Manishearth I considered it, but I think it has better chances of being answered here than on math.SE… If I don't get an answer here, I’ll follow your suggestion. – F'x May 20 '12 at 18:42
• I'm not a master of group theory or crystals in general either, and I had a hard time attempting to work it out from the linked descriptions, but handily, there's a site which will tell you. – Aesin May 22 '12 at 16:38

## 1 Answer

The Pmma is a minimal non-isomorphic supergroup of Pmm2; or you can say that Pmm2 is one of maximal non-isomorphic sub-groups of Pmma.

How to solve the question:

(1) fast and dirty: take International Tables for Crystallography vol. A go to your group (in mine edition Pmma starts on p.274). On second page describing space group you can find Maximal non-isomorphic subgroups. The third listed is  Pmm2 (25) 1; 2; 7; 8.

You can see that the operation in #50 and #25 have a lot of similarities. No.1 you can ignore as it's [x,y,z] but 2, 7 and 8 for #50 are [-x+.5,-y,z], [x,-y,z] and [-x+.5,y,z]. As one can see the #25 has following general operators: (2) [-x,-y,z] (3) [x,-y,z] and (4) [-x,y,z].

(2) The hard way: Mathematics is explain in the same volume in chapter 13.1. Isomorphic subgroups. Of course the knowledge of group theory and matrix calculus is to your advantage ;-)