Talk:Prüfer group

Wrong definition ?

The article gives this definition : "the Prüfer p-group or the p-quasicyclic group or p^∞-group, Z(p^∞), for a prime number p is the unique torsion group in which every element has p pth roots." I think this definition is wrong. The multiplicative group of all complex roots of unity is a torsion group in which every element has exactly p pth roots, but this group is not isomorphic to the Prüfer p-group. Marvoir (talk) 08:46, 1 December 2010 (UTC)[reply]

Yes, you're right. I changed "torsion group" to "p-group".82.35.82.162 (talk) 01:52, 23 January 2011 (UTC)[reply]

Yes, you have to eliminate the possibility of elements whose order is not a power of p. Good catch. -Krasnoludek (talk) 14:49, 11 February 2011 (UTC)[reply]

The Prüfer p-group cannot be a ring

The article reports "As both the integers $\mathbb {Z}$ and the p-adic rationals $\mathbb {Z} [1/p]$ are rings in addition to groups, the quotient ring $\mathbb {Z} (p^{\infty })=\mathbb {Z} [1/p]/\mathbb {Z}$ is the Prüfer p-group with a ring structure, or the Prüfer p-ring. "

First things first, no reference is given and cannot be found (there is a notion of Prüfer ring/domain, but that's a different thing). About the quote:

1) The quotient $\mathbb {Z} (p^{\infty })=\mathbb {Z} [1/p]/\mathbb {Z}$ is not a quotient ring because $\mathbb {Z}$ is not an ideal. However, it makes sense as a quotient of additive groups (and it is the Prüfer p-group, indeed).

2) The Prüfer p-group does not admit a ring structure in either cases in which the group multiplication plays the rôle of ring addition (it cannot be of finite characteristic since every element has arbitrarily large additive order, but every element is of finite additive order, hence none can be the ring unit) or multiplication (see https://math.stackexchange.com/a/93411). 06:37, 21 April 2020‎ 188.152.82.187

The stack exchange post just says that this cannot be a ring with identity. It can still be a ring, and

\mathbb {Z} [1/p]/\mathbb {Z}

with its usual multiplication seems to satisfy the ring axioms. — Preceding unsigned comment added by 24.250.184.73 (talk) 18:28, 21 October 2020 (UTC)[reply]

ℚ/ℤ

The quotient group ℚ/ℤ is the direct sum of the Prüfer groups Z(p^∞) over all primes p.

This fact certainly seems important enough to mention in this article.

I hope someone knowledgeable about this subject will add this fact.