What is Abelian group: Definition and 58 Discussions

In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel.The concept of an abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood and fully classified.

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  1. G

    Subatomic rotations in a plane Abelian group

    Homework Statement 5. The Z boson has a width of 2.4952 GeV: a. The Z decays 3.363% of the time in e+e-calculate the partial width of Z \rightarrow e+e-. b. The J/ \psi (A cc bar state) has a width of 93.4 KeV. Is its lifetime is longer or shorter than the Z lifetime? Explain. c...
  2. P

    What are presentations in group theory?

    Homework Statement G=(Z+Z+Z)/N where Z denote the integers and + is direct sum and N = <(7,8,9), (4,5,6), (1,2,3)> or the smallest submodule of Z+Z+Z containing these 3 vectors. How would you describe G? The Attempt at a Solution N = {a(7,8,9)+b(4,5,6)+c(1,2,3)|a,b,c in Z} = {(7a+4b+c...
  3. R

    How Do You Prove a Group is Abelian?

    Please HELP! So, I have to go about proving the following, but I have no idea where to even start: I. Let S = R – {3}. Define a*b = a + b – (ab)/3. 1. Show < S,*> is a binary operation [show closure]. 2. Show < S,*> is a group. 3. Find *-inverse of 11/5 II. Let G be a group with x,y...
  4. G

    Proving Subgroups of Free Abelian Groups: A Troubleshooting Guide

    I'm working on a proof for subgroups of free abelian groups and am having trouble with a step (I know other methods, but would like to try and make this one work if possible). The basic idea is let G be a free abelian group with generators (g_1...g_n) and let H be a subgroup of G. Assuming a...
  5. T

    Proving Abelian Group with Numbers and Operations

    Hello. I was wondering how I could prove if a set of numbers along with some arbitrary operation is an abelian group.
  6. C

    Counterexample involving an abelian group

    Basically, I have to show an example such that for a nonabelian group G, with a,b elements of G, (a has order n, and b has order m), it is not necessarily the case that (ab)^mn= e. where e is the identity element. im not sure where to start. =\
  7. B

    Abelian group isomorphism help

    let G be an abelian group, and n positive integer phi is a map frm G to G sending x->x^n phi is a homomorphism show that a.)ker phi={g from G, |g| divides n} b.) phi is an isomorphism if n is relatively primes to |G| i have no clue how to even start the prob...:-(
  8. Nebula

    Question: Elements of Order 2 in Finite Abelian Group

    I've got a question. It pertains to a proof I'm doing. I ran into this stumbling block. If I could show this I think I could complete the proof. G is a finite Abelian Group such that there exits more than one element of order 2 within the group. more than one element of the form b not...
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