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Let $P$ be a point group of a crystallographic group $G = (P, \mathbb{Z}^n, \alpha)$, and $N$ be its normalizer, i.e. for every $A \in N$ we get $AP=PA$. I'm interested in the structure of $N$. For dim $3$, I have constructed the normalizer for every point group and noticed that either:

  • $N = GL_n(\mathbb{Q})$
  • every matrix of $N$ has permutational form, e.g. $$\left(\begin{array}{rrr} 0 & x_0 & 0 \\ x_1 & 0 & 0 \\ 0 & 0 & x_2 \end{array}\right)$$ or is decomposable, e.g. $$\left(\begin{array}{rrr} -x_0 & x_0 + x_1 & 0 \\ x_1 & x_0 & 0 \\ 0 & 0 & x_2 \end{array}\right).$$

Moreover, the diagonal block of decomposable matrices coincides with the normalizer of some point group of dim 2.

Question: Why is this true? Can someone provide a reference? Is it true for higher dimensions?

Context: I'm interested in characteristic polynomials of matrices from the normalizer, particularly when they are divisible by a monic polynomial with integral coefficients. In other words, I wonder when the operator $A \in N$ has an invariant sublattice $L' < \mathbb{Z}^n$. The observation in question can reduce the problem to the lower dimensions.

davendiy
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  • For this question to be answerable, you have to specify the ambient group, the space it acts on and the action. You provided neither. – Moishe Kohan Aug 15 '25 at 14:10
  • The point group $P$ used in the context of crystallographic groups, acting on $\mathbb{R}^n$. The normalizer $N$ consists of rational matrices $A$ such that $PA = AP$. The matrices used in the examples were obtained by solving this system of linear equations $PA=AP$ over rationals. – davendiy Aug 17 '25 at 10:15
  • I can only repeat my earlier suggestion. With the definition I know, I do not see how all rational matrices are relevant. Instead, one should take a crystallographic group $\Gamma$ and consider all elements $\gamma\in \Gamma$ such that $\gamma P\gamma^{-1}=P$. Furthermore, the definition depends on the fixed point of the group $P$ ($P$ is the isotropy group of a specific point in $\mathbb R^n$). I suggest, you start by writing down carefully all the relevant definitions. – Moishe Kohan Aug 17 '25 at 13:23
  • @MoisheKohan, I'm interested in virtual endomorphisms of crystallographic groups as abstract groups, i.e. endomorphisms $\phi : H \rightarrow G$ where $H$ is a subgroup of finite index of $G$. Particularly, consider the case when $H$ is isomorphic to $G$. Then $H = a^{-1}Ga$ for some affine $a = (A, t)$. To ensure that conjugation holds elements in $G$ we at least need $AP = PA$, interpreting $P$ as a finite group of nxn matrices (not n+1 x n+1 ones in homogeneous coordinates).

    Maybe there are some techniques for describing every possible index of isomorphic crystallographic subgroups?

     – davendiy Aug 18 '25 at 18:47
  • I think your post needs a complete revision, including the title. The question about virtual endomorphisms is very different from one about normalizers of isotropy subgroups. – Moishe Kohan Aug 18 '25 at 19:22

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