Timeline for Isometry group of an integer
Current License: CC BY-SA 3.0
7 events
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| Mar 5, 2018 at 8:50 | comment | added | Sylvain JULIEN | I guess you're right but still, it may give rise to interesting questions like détermining the sets $ \mathbb{N}_{G} : =\{n,G(n)\cong G\} $ that form a partition of $ N $ or the sequence of '$ G $ -type radii of $ n $' defined as the non negative integers $ r $ such that $ G(n-r)\cong G(n+r)\cong G $ . Of course $ \mathbb{N}_{\mathbb{Z}/2\mathbb{Z}} $ is the set of prime numbers. | |
| Mar 5, 2018 at 8:10 | comment | added | Greg Martin | Great point, showing I need to recalibrate. Ok, if $n=\prod_{i=1}^k p_i^{a_i}$, then the parallelotope is the direct product of $k$ hypercubes, where the $i$th hypercube has dimension $a_i$. Therefore $G(n)$ contains the product of the isometry groups of these hypercubes. (Even a $1$-dimensional hypercube has a nontrivial isometry, I should have realized.) Again the question to consider is: are there ever any additional isometries? for if not, it again boils down to integers with a prescribed factorization type. | |
| Mar 5, 2018 at 6:28 | comment | added | Sylvain JULIEN | @Greg Martin : it seems to me that the involution you consider is geometrically the reflexion with respect to a hyperplane that contains the diagonal of the square with size 2, so yes there are other elements : in your example the dihedral group of order 8 is a subgroup of $ G(n) $ . | |
| Mar 5, 2018 at 1:27 | comment | added | Greg Martin | Obviously $G(n)$ contains an appropriate group of coordinate-permutation isometries; for example, if $n=4m$ where $m$ is odd and squarefree, then $G(n)$ includes the involution that exchanges the two coordinates where $a_i$ has norm $2$ and fixes the other coordinates. Do you have an example where $G(n)$ is strictly larger than this trivial subgroup? If there are no such examples, then you're just asking about the counting function of integers with a fixed factorization type, which is an easy question. | |
| Mar 5, 2018 at 0:16 | comment | added | Adam P. Goucher | Do you mean the direct sum of hyperoctahedral groups, $\bigoplus_{i \in I} B_{a_i}$? | |
| Mar 4, 2018 at 23:12 | history | edited | Sylvain JULIEN | CC BY-SA 3.0 |
added 121 characters in body
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| Mar 4, 2018 at 22:48 | history | asked | Sylvain JULIEN | CC BY-SA 3.0 |