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. You can find more information on torsion subgroups at this sci.math posting: Re: structure of arbitrary Abelian groups - Dave Rusin http://www.math.niu.edu/~rusin/known-math/99/ab_gps I hope this helped answer the questions you had concerning your mathematics problem. - Doctor Nitrogen, The Math Forum http://mathforum.org/dr.math/ |
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