Subgroups of crystallographic groups

Asked Feb 20 '23 at 17:09

Active Feb 20 '23 at 18:31

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I am trying to understand the structure of crystallographic groups and came to the following question. Let $G_1$ and $G_2$ be two non-isomorphic crystallographic groups of the same dimension and the same point groups. Can $G_1$ contain a subgroup isomorphic to $G_2$? In other words, can $G_2$ be a subgroup of $G_1$?

I tried to play with 1-cocycles, but I don't see how a 1-cocycle could change when we pass to a subgroup. Also I tried to conjugate one group inside another group -- came to nothing.

edited Feb 20 '23 at 18:30

asked Feb 20 '23 at 17:09

QMath

So we have extensions $1\rightarrow \Bbb Z^n\rightarrow G_i\rightarrow F_i\rightarrow 1$ for $i=1,2$, and $F_1\cong F_2$, $G_2\le G_1$ but $G_1\not\cong G_2$? – Dietrich Burde Feb 20 '23 at 19:44
Yes, is it possible? #Comments must be at least ... – QMath Feb 20 '23 at 19:58
I have found the answer. It is possible even in dimension 2. – QMath Feb 21 '23 at 08:39
Which of the $17$ groups did you take for $G_1$ and $G_2$ in dimension $2$? See here for a reference with extensions. – Dietrich Burde Feb 21 '23 at 08:53
The group p1m1 (No. 3) contains as a subgroup the group p1g1 (No. 4), while both groups have $C_2$ at their point groups. – QMath Feb 21 '23 at 11:26
1

I see. The notes by Morandi have some details using cohomology, also for p1m1 and p1g1 (the point group is $D_{1,p}\cong C_2$), with cocycles. Perhaps it is interesting for you. – Dietrich Burde Feb 21 '23 at 12:17

Subgroups of crystallographic groups

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