{"id":986,"date":"2014-09-15T01:00:17","date_gmt":"2014-09-15T01:00:17","guid":{"rendered":"http:\/\/blogs.ams.org\/visualinsight\/?p=986"},"modified":"2018-02-08T23:14:33","modified_gmt":"2018-02-08T23:14:33","slug":"prufer-2-group","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/visualinsight\/2014\/09\/15\/prufer-2-group\/","title":{"rendered":"Pr\u00fcfer 2-Group"},"content":{"rendered":"
\n
\"Pr\u00fcfer<\/a>

Pr\u00fcfer 2-group<\/p><\/div>\n<\/div>\n

This is the Prüfer $2$-group<\/b>, the subgroup of the unit complex numbers consisting of all $2^n$th roots of unity. It is also called $\\mathbb{Z}(2^\\infty)$. It is generated by elements <\/p>\n

$$\\begin{array}{l} g_1 = e^{i \\pi} = -1 , \\\\
\n g_2 = e^{i \\pi\/2} = i, \\\\
\n g_3 = e^{i \\pi\/4}, \\\\
\n g_4 = e^{i \\pi \/ 8}, \\dots \\end{array} $$<\/p>\n

with relations <\/p>\n

$$ g_{n+1}^2 = g_n , \\quad g_1^2 = 1 $$<\/p>\n

In the picture, elements are labelled by their names in terms of these generators.<\/p>\n

In general, for any prime $p$ the Prüfer $p$-group<\/a><\/b> is the subgroup of the unit circle consisting of all $p^n$th roots of unity:<\/p>\n

$$ \\mathbb{Z}(p^\\infty) = \\{e^{2 \\pi i m \/ p^n} : m, n = 1,2,3, \\dots \\} $$<\/p>\n

We can also think of the Prüfer $p$-group as a subgroup of $\\mathbb{Q}\/\\mathbb{Z}$, namely the group consisting of all equivalence classes of fractions where the denominator is a power of $p$.<\/p>\n

More abstractly, we can think of the Prüfer $p$-group as the colimit of the groups <\/p>\n

$$ \\mathbb{Z}\/p \\mathbb{Z} \\rightarrow \\mathbb{Z}\/p^2 \\mathbb{Z} \\rightarrow \\mathbb{Z}\/p^3 \\mathbb{Z} \\rightarrow \\cdots $$<\/p>\n

As a $\\mathbf{Z}$-module, the Pr\u00fcfer $p$-group is Artinian<\/b><\/a>. This means that that any infinite descending sequence of subgroups of the Pr\u00fcfer $p$-group ‘bottoms out’ after finitely many steps. On the other hand, it is not Noetherian<\/b><\/a>: there are infinite ascending sequences of subgroups that keep getting bigger forever. So, it is a good example to distinguish the concepts of ‘Artinian’ and ‘Noetherian’.<\/p>\n

The Prüfer $p$-group can be given a topology induced from the usual topology on the circle, but it can also be given the discrete topology. With its discrete topology, it is the Pontryagin dual<\/a> of the compact abelian group of $p$-adic integers<\/a>. We will explore this more in the next article.<\/p>\n

This image is from Wikicommons<\/a>, where it was placed into the public domain by its creator, known only as Anon213487. I have no idea why the unit circle is drawn as an ellipse here, but it’s sort of cute.<\/p>\n

\n

Visual Insight<\/i> is a place to share striking images that help explain advanced topics in mathematics. I\u2019m always looking for truly beautiful images, so if you know about one, please drop a comment here<\/a> and let me know!<\/p>\n

<\/div>","protected":false},"excerpt":{"rendered":"

This is the Prüfer $2$-group<\/b>, the subgroup of the unit complex numbers consisting of all $2^n$th roots of unity. It is also called $\\mathbb{Z}(2^\\infty)$. <\/p>\n

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