English: The
compact group of
3-adic integers (black points), with selected elements labeled by the corresponding character on the
Pontryagin dual group (the
discrete Prüfer 3-group) (colored discs).
Counter-clockwise from the right, the labeled elements are 0, 9, 3, 1, −1/8, −1/2, 1/4, −1/4, 1/2, 1/8, −1, −3, and −9. Open the SVG directly in your browser to get tooltips on group elements.
Each colored disc is tied to a 3-adic integer,
, and it represents the corresponding character on the Prüfer 3-group,
, defined by
. The circle group
used is a color wheel where 0 = red, 1/3 = green, and 2/3 = blue.
For details on the embedding of the 3-adic integers, see Chistyakov, D. V. (1996), “Fractal geometry for images of continuous embeddings of
p-adic numbers and
p-adic solenoids into Euclidean spaces”, in
Theoretical and Mathematical Physics, volume 109, issue 3,
DOI:10.1007/BF02073866, pages 1495–1507. The particular mapping used is
![{\displaystyle \Upsilon _{s}^{(\infty )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c719b6e9a5179d2fa21fefe0e131dec3f9c71235)
, defined in Definition 3 and depicted in Figure 1.12.