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.

View More On Wikipedia.org
Recent contents Top users
  1. Lauren1234

    Is this matrix a non-abelian group?

    I know for a group to be abelian a*b=b*a I tried multiplying the matrix by itself also but I’m not sure what I’m looking for. picture is below of the matrix https://www.physicsforums.com/attachments/255812
  2. A

    I How can there only be two possible four-element groups?

    How can you prove that there can only be 2 possible four-element group?
  3. Konte

    I Can an Abelian Group Be Isomorphic to a Non-Abelian Group in Physics?

    Hi everybody, I have a question: is an abelian group can be isomorphic to a non-abelian group? Thank you everybody.
  4. Mr Davis 97

    Proving that an Abelian group of order pq is isomorphic to Z_pq

    Homework Statement Given that G is an abelian group of order pq, I need to show that G is isomorphic to ##\mathbb{Z}_{pq}## Homework EquationsThe Attempt at a Solution I am trying to do this by showing that G is always cyclic, and hence that isomorphism holds. If there is an element of order...
  5. Mr Davis 97

    Abelian group as a direct product of cyclic groups

    Homework Statement Consider G = {1, 8, 12, 14, 18, 21, 27, 31, 34, 38, 44, 47, 51, 53, 57, 64} with the operation being multiplication mod 65. By the classification of finite abelian groups, this is isomorphic to a direct product of cyclic groups. Which direct product? Homework EquationsThe...
  6. alexmahone

    MHB Show that G is an Abelian group

    Let $G$ be a group such that for all $a$, $b$, $c$, $d$, and $y\in G$ if $ayb=cyd$ then $ab=cd$. Show that $G$ is an Abelian group. HINTS ONLY as this is an assignment problem.
  7. G

    MHB Prove Abelian Group: $(a \cdot b)^n = a^n \cdot b^n$

    Prove that if $G$ is an abelian group, then for $a, b \in G$ and all integers $n$, $(a \cdot b)^n = a^n \cdot b^n.$ Never mind. I figured it out. We proceed by induction on $n$, then use a lemma in the text.
  8. A

    Abstract Algebra: Abelian group order

    Homework Statement Let G be an abelian group and let x, y be elements in G. Suppose that x and y are of finite order. Show that xy is of finite order and that, in fact, o(xy) divides o(x)o(y). Assume in addition that (o(x),(o(y)) = 1. Prove that o(xy) = o(x)o(y). The Attempt at a...
  9. V

    Supose that G is a finite abelian group that does not contain a subgro

    let us assume G is not cyclic. Let a be an element of G of maximal order. Since G is not cyclic we have <a>≠G. Let b be an element in G, but not in the cyclic subgroup generated by a. O(a) = m and O(b) = n where O() refers tothe orders. . then how can we use this to construct a subgroup of G...
  10. D

    Why does an abelian group of order G have G conjugacy classes

    Homework Statement Hi guys, The title pretty much says it. I need to explain why: (a) an abelian group of order |G| has precisely |G| conjugacy classes, and (b) why the irreducible representations of abelian groups are one-dimensional. Also in my description below, if I make any mathematical...
  11. caffeinemachine

    MHB Finite abelian group textbook help

    I need to read about finite abelian groups. I searched 'finite abelian group' on amazon and the closest search result was 'finite group theory'. Googling didn't help either. Does there exist a book dedicated to finite abelian groups? If yes, and if you know of a good one then please reply...
  12. R

    Proving Finite Order Elements Form a Subgroup of an Abelian Group

    Homework Statement Prove the collection of all finite order elements in an abelian group, G, is a subgroup of G. The Attempt at a Solution Let H={x\inG : x is finite} with a,b \inH. Then a^{n}=e and b^{m}=e for some n,m. And b^{-1}\inH. (Can I just say this?) Hence...
  13. J

    Subgroup of Finitely Generated Abelian Group

    Homework Statement Prove that any subgroup of a finitely generated abelian group is finitely generated. Homework Equations The Attempt at a Solution I've attempted a proof by induction on the number of generators. The case n=1 corresponds to a cyclic group, and any subgroup of a...
  14. D

    Proving the Identity Property in Abelian Groups

    Hi guys, I have quastion about groups: G is abelian group with an identity element "e". If xx=e then x=e. Is it true or false? I was thinking and my feeling is that it's true but I just can't prove it. I started with: (*) ae=ea=a (*) aa^-1 = a^-1 a = e those from the...
  15. QuestForInsight

    MHB What is an Abelian group and why is it useful in mathematics?

    Let $\mathbb{G}$ be a set with a map $(\xi, ~ \eta) \mapsto f(\xi, ~\eta)$ from $\mathbb{G}\times\mathbb{G}$ into $\mathbb{G}$. For every pair $(\xi, ~ \eta)$ in $\mathbb{G}$ let $f(\xi, ~\eta) = f(\eta, ~ \xi)$. Suppose there are elements $\omega$ and $\xi'$ in $\mathbb{G}$ such that for every...
  16. T

    Find a direct summand of a finite abelian group

    Homework Statement If G is a finite abelian group, and x is an element of maximal order, then <x> is a direct summand of G. Homework Equations The Attempt at a Solution I claim that the hypothesis implies that A = G\<x> \bigcup {e} is a subgroup of G. If so, then since G = < <x> \bigcup A>...
  17. N

    Abelian group with order product of primes = cyclic?

    It seems rather straight forward that if you have an abelian group G with \# G = p_1 p_2 \cdots p_n (these being different primes), that it is cyclic. The reason being that you have elements g_1, g_2, \cdots g_n with the respective prime order (Cauchy's theorem) and their product will have to...
  18. I

    Can prime fields act two ways on the same abelian group?

    A problem asks to find an abelian group V and a field F such that there exist two different actions, call them \cdot and \odot, of F on V such that V is an F-module. A usual way to solve this is to take any vector space over a field with a non-trivial automorphism group, and define r\odot \mu...
  19. J

    Abelian group on the natural numbers (including 0) ?

    Is it possible to define an abelian group on the natural numbers (including 0)? It's just that for every binary operation I've tried, I can't find an inverse!
  20. D

    Proving Non-Abelian Groups Have Unique Elements with Non-Commutative Properties

    To show that a non-abelian group G, has elements x,y,z such that xy = yz where y≠z, Is it enough to simply state for non-abelian groups xy≠yx so if you have xy=yz then it is not possible for x=z due to xy≠yx? Or is more detail required?
  21. J

    Finite abelian group into sequence of subgroups

    G finite abelian group WTS: There exist sequence of subgroups {e} = Hr c ... c H1 c G such that Hi/Hi+1 is cyclic of prime order for all i. My original thought was to create Hi+1 by reducing the power of one of the generators of Hi by a prime p. Then the order of Hi/Hi+1 would be p, but...
  22. T

    Show abelian group has element of order [m,n]

    Let G be an abelian group containing elements a and b of order m and n, respectively. Show that G contains an element of order [m,n] (the LCM of m and n). This is true when (m,n)=1, because mn(a+b) = e, and if |a+b|=h, then h|mn. Now, hm(a+b) →m|h and similarly I find that n|h. But (m,n)=1...
  23. H

    Order of subgroup of an abelian group

    Homework Statement Suppose H and K are subgroups of an abelian group G (not neccessarily finite). Let the order of H and K be a and b respectively. Prove that there exists a subgroup of order L, where L = lcm(a,b). Homework Equations Product Formula: |HK|/|H| = |K|/|H intersect K|...
  24. A

    What is the order of an element in a finite abelian group of odd order?

    Homework Statement Let G be a finite abelian group of odd order. Prove that the product of all the elements of G is the identity. The Attempt at a Solution easy to see the case when each element has inverse which is not itself.
  25. P

    Proof Abelian Group: Consec Ints & No Follow w/2

    Homework Statement If G is a group in which (a.b)^i=a^i.b^i for three consecutive integers i for all a,b \in G , show that G is abelian. Show that the conclusion does not follow if we assume the relation (a.b)^i=a^i.b^i for just two consecutive integers. Homework Equations The...
  26. D

    Abelian Group Order: Find Largest Possible # of Subgroups

    This isn't homework, it was proposed by a professor of mine and I'm dying here because the hint makes no sense to me Let G be an abelian group of order 540, what is the largest possible number of subgroups of order 3 such a group G can have? He said to classify abelian groups of order 27...
  27. B

    Subgroups of a finitely generated abelian group

    I have a homework problem which asks to prove that the subgroups of a finitely generated abelian group are finitely generated. The hint in the book says to prove it by induction on the size of X where the group G = <X>. It also says to consider the quotient group G/an+1 (with an+1 in X) in the...
  28. 1

    Proving an Abelian Group with 2 Elements of Order 2 has a Subgroup of Order 4

    Homework Statement Prove that an abelian group with two elements of order 2 must have a subgroup of order 4 Homework Equations The Attempt at a Solution Let G be an abelian group ==> for every a,b that belong to G ab=ba. Let a,b have order 2 ==> a^2 =e and b^2 = e. Since a...
  29. S

    Subgroup of given order of an Abelian group

    Hey! We know that if there exists an element of a given order in a group, there also exists a cyclic subgroup of that order. What about converse? Suppose there is a subgroup of an Abelian group of order 'm'. Does that imply there also exists an element of order 'm' in the Group. It does not...
  30. F

    Finite Abelian Group: Proving # Subgps Order p = # Subgps Index p

    Homework Statement A be a finite abelian group, prove # of subgps of order p = # of subgps of index p, p is a prime. The Attempt at a Solution I have thought about this probably very easy problem for 2 hours and could not find a satisfying proof. I have tried bijective proof but...
  31. N

    Finite Abelian Group: Showing Isomorphism & Subgroups of Order/Index p

    Suppose A is a finite abelian group and p is a prime. A^p={a^p : a in A} and A_p={x:x^p=1,x in A}. How to show A/A^p is isomorphic to A_p. I tried to define a p-power map between A/A^p and A_p and show this map is isomorphism. But my idea didnot work right now. Please give me some help. In...
  32. G

    Cyclic abelian group of order pq

    I'm looking at the exercises of Hungerfod's Algebra. Some looks easy but it seems the proofs are not so obvious. Here's one I'm particularly having a hard time solving: Let G be an abelian group of order pq with (p,q)=1. Assume that there exists elements a and b in G such that |a|= p and |b|...
  33. C

    Proving H is Cyclic: Finite Abelian Group

    Homework Statement Let H be a finite abelian group that has one subgroup of order d for every positive divisor d of the order of H. Prove that H is cyclicHomework Equations We want to show H={a^n|n is an integer}
  34. K

    Abelian Group Proof: Example to Show Necessity

    Homework Statement If G is an abelian group, then (ab)^2=a^2b^2 for all a, b in G. Give an example to show that abelian is necessary in the statement of the theorem. Homework Equations The Attempt at a Solution Abelian implies commutativity. a*b=?b*a a^2b^2=b^2a^2 For example...
  35. S

    Finite abelian group of size p-1

    Suppose we have two finite abelian groups G,G^{\prime} of size n=pq, p,q being primes. G is cyclic. Both G,G^{\prime} have subgroups H,H^{\prime}, both of size q. The factor groups G/H,\ G^{\prime}/H^{\prime} are cyclic and since they are of equal size, they are isomorphic. Are G,G^{\prime}...
  36. T

    Free Abelian Group Homomorphisms into S_3 - Count & Answer

    Homework Statement How many different homomorphisms are there of a free abelian group of rank 2 into S_{3}? Where S_{3} is the symmetric group of 3 elements. The Attempt at a Solution I think 12 but the answers suggest 18. ?
  37. A

    Set R^(2) with the usual vector addition forms an abelian group

    Homework Statement the set R^(2) with the usual vector addition forms an abelian group. For a belongs to R and x=(x1,x2) belongs to R^(2) we put a *x :=(ax1,0),this defines a scalar multiplication R*R^2 ---R^2 (a,x)---a*x. determine which of the axioms defining a vector space hold for the...
  38. D

    Cyclic subgroups of an Abelian group

    Homework Statement If G is an Abelian group and contains cyclic subgroups of orders 4 and 6, what other sizes of cyclic subgroups must G contain? Homework Equations A cyclic group of order n has cyclic subgroups with orders corresponding to all of n's divisors. The Attempt at a...
  39. J

    Additive abelian group where x+x+x+x=0

    (Problem 49 from practice GRE Math exam:) Up to isomorphism, how many additive abelian groups G of order 16 have the property that x + x + x + x = 0 for each x in G? (A) 0 (B) 1 (C) 2 (D) 3 (E) 5 The answer is (D) 3, but I don't understand what the problem is asking, really, and I don't...
  40. P

    Least common multiple of the orders of the elements in a finite Abelian group

    Homework Statement (From an exercise-section in a chapter on Lagrange's theorem:) Let G be a finite abelian group and let m be the least common multiple of the order of its elements. Prove that G contains an element of order m. The Attempt at a Solution We have x^m = e for all x in G. By...
  41. U

    Abelian group w/ 1000 elements?

    Abelian group w/ 1000 elements? My guess is... Z mod 100 under addition. I am new to algebra though...any thoughts? Also, what about multiplication?
  42. E

    Two Finite Abelian Group Problems

    Homework Statement 1. Suppose that G is a finite Abelian group that has exactly one subgroup for each divisor of |G|. Show that G is cyclic. 2. Suppose that G is a finite Abelian group. Prove that G has order pn, where p is prime, if and only if the order of every element of G is a power...
  43. E

    Is Every Group with Squared Elements Equal to the Identity Element Abelian?

    [SOLVED] An Abelian Group Problem Homework Statement Prove: If G is a group where the square of every element equals the identity element, then G is Abelian. The attempt at a solution I've been able to prove is that a-1 = a and that (ab)-1 = ba where a and b are in G. Everything else I've...
  44. H

    Unique abelian group of order n

    Homework Statement Determine all integers for which there exists a unique abelian group of order n. Homework Equations The Attempt at a Solution All prime integers?
  45. E

    Homomorphisms from a free abelian group

    Homework Statement How many homomorphism are there of a free abelian group of rank 2 into a) Z_6 and b) S_3.Homework Equations The Attempt at a Solution Since the images of the generators completely determine a homomorphism, the upper bound for both is 36. Now a free abelian group of rank 2 is...
  46. L

    Abelian Group; what to do if the set is G=R-{1/3}?

    Homework Statement On the set G=R-{1/3} the following operation is defined: *G: GxG arrow G (x,y) arrow x*y=x+y-3xy Show that (G,*) is an abelian group. Homework Equations To proove something is an abelian group: The Associative Law need to hold true x*(y*x)=(x*y)*x...
  47. K

    Is a Group Abelian if a*a=e?

    Homework Statement Let G be a group such that a*a = e for all a\inG. Show that G is abelian. Homework Equations The Attempt at a Solution I know the conditions for an abelian group but don't see or understand the significance that a*a=e here?
  48. K

    Complex numbers as an abelian group

    [SOLVED] Complex numbers as an abelian group Homework Statement Multiplication of complex numbers defines a binary operation on C^x:=C\{0} (complex numbers not including zero). Show that C^x together with this binary operation is an abelian group. (without further discussion you may use the...
  49. E

    Solved: Free Abelian Group Rank R Example

    [SOLVED] free abelian group Homework Statement Show by example that is is possible for a proper subgroup of a free abelian group of finite rank r also to have rank r.Homework Equations The Attempt at a Solution I believe that there are no example in the set of finitely-generated free abelian...
  50. G

    Subatomic rotations in a plane Abelian group

    Hi.. I recently stumbled across a question that seemed a little bit odd "Show that the set of rotations in a plane form a SO(2) Abelian group." for a subatomic physics course. I know how to obtain the answer showing that A^TA=AA^T=1... what I don't understand is what the relevance to subatomic...
Back
Top