Talk:Torsion group

Aren't all finite groups periodic?

OK, I might have forgotten all my maths, but aren't all finite groups periodic? (Because every element has order at most the order of the group) This would make the 2nd sentence of the 1st para ("Finite cyclic groups are examples of periodic groups.") misleading - all finite groups, cyclic or not, are periodic. — Preceding unsigned comment added by 86.6.15.46 (talk) 20:42, 27 March 2006 (UTC)

Good point, my original edit was to try and tidy up the sentence

The concept of a periodic group should not be confused with that of a cyclic group.

which I find unsatisfactory. Somehow it seems it cyclic groups should get a mention, but I'm not sure how. --Salix alba (talk) 07:59, 28 March 2006 (UTC)

Example needed

There should be an example of periodic group which is not finite. --193.198.16.211 (talk) 21:15, 5 May 2008 (UTC)

Done. Nonabelian examples can be given in similar fashions: an infinite dimensional extraspecial group, the union of the finite symmetric groups, etc. JackSchmidt (talk) 14:03, 6 May 2008 (UTC)

please correct these links

In this section External links periodic groups and exponent have no correct links! — Preceding unsigned comment added by 188.210.75.253 (talk) 07:19, 15 May 2018 (UTC)

Done.—Anita5192 (talk) 14:34, 15 May 2018 (UTC)

@Anita5192 you could replace correct links as: http://planetmath.org/periodicgroup and http://planetmath.org/exponent respectively!

Verification of info from the cited book

I think that the accuracy of citation of info from the cited book by Ebbinghaus should be verified. It is claimed in a section that the use of an infinite disjunction is inadmissible in first order logic. Some sentences from the opening of the book say rather something else :

For example, the first-order language does not allow the formulation of an adequate axiom system for arithmetic or analysis. On the other hand, this difficulty can be overcome--even in the framework of first-order logic-by developing mathematics in set-theoretic terms.

cited by --93.122.249.165 (talk) 00:35, 29 August 2019 (UTC).

I (Vlad P.) am also confused by the way this "logical" formula is expressed. What exactly can stop us from, instead of an "infinite disjunction", writing this formula:

$\forall x.\exists n>=0.(x^{n}=e)$

Of course natural numbers are not a part of group theory, but people tend to think of group theory in terms of some set theory anyway, and sets usually do have natural numbers.

Vlad Patryshev (talk) 06:13, 21 February 2020 (UTC)

When you quantify $\exists n>=0.(x^{n}=e)$ you are quantifying over a formula $(x^{n}=e)$. To see this, write $\exists f\in F.(f(x)=e)$ where f is some formula, and F is the set of allowed formulas. Such quantification over formulas is allowed only in second-order logic, not FOL. It just so happens that, for you, it seems "simple" because $f(x)=x^{n}$ and F has countably many formulas in it, which can be identified with the integers. This makes it look deceptively simple. But they are still formulas. It would be nice if there was first-and-a-half-order logic, that allowed such deceptively simple formulas. But I suppose there is some reason why such a beast does not exist. But I don't know what that reason is. 67.198.37.16 (talk) 05:58, 11 June 2023 (UTC)