Associated Topics || Dr. Math Home || Search Dr. Math

### What is a Torsion Subgroup?

```Date: 03/05/2003 at 12:42:00
From: Kris
Subject: Torsion subgroup

Let G be an abelian group. Show that the elements of finite order in G
form a subgroup. This subgroup is called the torsion subgroup of G.
Now find the torsion subgroup of the multiplicative group R* of
nonzero real numbers.
```

```
Date: 04/04/2003 at 19:30:42
From: Doctor Nitrogen
Subject: Re: Torsion subgroup

Hi, Kris:

A "torsion subgroup" of an Abelian group G is a subgroup of elements
from G all having finite order for some positive integer m. It is
known that if G is Abelian and T is a torsion subgroup of G, then the
factor group

G/T

is torsion free.

Torsion subgroups are studied somewhat in R-module and L-module
theory, analogues of a kind of a vector space over a field, where
instead of having a vector space over a field, you have "a group over
a Ring." They also have some connection to elliptic curves in number
theory.

Concerning your question, I'll do part of it for you and you can
reason out the rest.

Let T be a subset of elements of finite order m in G. Let "e" denote
the identity. Then

(1) e is an element of T, so T has an identity.

(2) Since G is an Abelian group which is associative, T is also
associative (and Abelian) since it is a subset of G.

(3)For "a" any element of T,

a^m

= a(a^m-1)

= (a^m-1)(a)

= (a^-1)(a)

= e

Hence every element in T has an inverse.

(4)For a an element of T,

(ae)

= (ea)

= a

Hence T satisfies all the requirements for it to be a subgroup of G.

I will leave the rest of the assignment for you to ponder.

posting:

Re: structure of arbitrary Abelian groups - Dave Rusin
http://www.math.niu.edu/~rusin/known-math/99/ab_gps

mathematics problem.

- Doctor Nitrogen, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Modern Algebra

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search