1. Every subgroup of an abelian group is abelian.

2. Every subgroup of a free abelian group is itself a free abelian group.

3. A torsion abelian group is an abelian group in which every element has finite order.

4. Every abelian group is a T-group.

5. Every cyclic group is abelian.

6. By the Fundamental Theorem of Finite Abelian Groups, every Abelian group of order 144 is isomorphic to the direct product of an Abelian group of order 16 = 24 and an Abelian group of

7. Every subgroup and factor group of a finitely generated abelian group is again finitely generated abelian.

8. Any group that is virtually abelian.

9. Abelian group 1 Abelian group In abstract algebra, 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 their order (the axiom of commutativity).

10. Abelian group: a group whose binary operation is commutative.

11. All cyclic groups are Abelian, but an Abelian group is not necessarily cyclic

12. Every abelian group can be embedded in a divisible group.

13. Abelian definition: of or relating to an Abelian group Meaning, pronunciation, translations and examples

14. Usually E is an additive abelian group.

15. In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian.

16. Abelian actually means that the group has commutativity

17. All subgroups of an Abelian group are normal.

18. For finite groups, being Abelian and the automorphism group being Abelian as well implies cyclic

19. However, every group of order p2 is abelian.

20. Any direct sum of finitely many finitely generated abelian groups is again a finitely generated abelian group.

21. The fundamental group of an H-space is abelian.

22. A finitely generated torsion-free abelian group is free.

23. The category of abelian groups is the fundamental example of an abelian category, and accordingly every subgroup of an abelian group is a normal subgroup.

24. An Abelian group is a group for which the elements commute (i.e., for all elements and). Abelian groups therefore correspond to groups with symmetric multiplication tables

25. The Cayley table tells us whether a group is abelian.

26. The Nielsen–Schreier theorem is a non-abelian analogue of an older result of Richard Dedekind, that every subgroup of a free abelian group is free abelian.

27. For example, the conjugacy classes of an Abelian group consist of singleton sets (sets containing one element), and every subgroup of an Abelian group is normal.

28. The smallest non-abelian group is the symmetric group S3 which has 3! = 6 elements.

29. Abelian synonyms, Abelian pronunciation, Abelian translation, English dictionary definition of Abelian

30. For instance, the additive group of rational numbers has an Abelian automorphism group (the multiplicative group of rational

31. Every abelian subgroup of a Gromov hyperbolic group is virtually cyclic.

32. The group F ab (S) is called the free Abelian group generated by the set S

33. Formally, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point O. An elliptic curve is an abelian variety – that is, it has a multiplication defined algebraically, with respect to which it is an abelian group – and O serves as the identity element.

34. Abelian groups 1 Definition An Abelian group is a set A with a binary operation satisfying the following conditions: (A1) For

35. The Brauer group of a finite extension of a quasi-algebraically closed field is trivial.

36. If 0=1, (1)-(3) are the axioms for an abelian group.

37. Abelian groups are generally simpler to analyze than nonAbelian groups are, as many objects of interest for a given group simplify to special cases when the group is Abelian

38. Definition of Abelian : commutative sense 2 Abelian group Abelian ring Examples of Abelian in a Sentence Recent Examples on the Web Microsoft is trying to chase a new quantum computer based on a new topography and a yet-undiscovered particle called non-Abelian anyons.

39. As these two prototypes are both abelian, so is any cyclic group.

40. Any algebraic extension of a quasi-algebraically closed field is quasi-algebraically closed.

41. A group that is not abelian but for which every subgroup is normal is called a Hamiltonian group.

42. Mathematically, QED is an abelian gauge theory with the symmetry group U(1).

43. A semisimple Lie group does not have any non-discrete normal abelian subgroups.

44. An Abelian variety is an algebraic group that is a complete algebraic variety

45. Abelian Integrals and Abelian Functions

46. In mathematics, the Mordell–Weil theorem states that for an abelian variety A over a number field K, the group A(K) of K-rational points of A is a finitely-generated abelian group, called the Mordell-Weil group.

47. Pertaining to an algebraic system in which an operation is commutative: an Abelian group

48. He also gave the first example of abelian varieties with finite Tate–Shafarevich group.

49. An Abelian group A A is a free Abelian group of rank r r if there exist u1,,ur ∈ A u 1,, u r ∈ A such that A = ⟨u1,,.ur⟩ A = ⟨ u 1,,

50. A definition of an Abelian group is provided along with examples using matrix groups

51. It can be used to measure how close to be abelian a finite group is.

52. The abelian group of translations is a normal subgroup, while the Lorentz group is also a subgroup, the stabilizer of the origin.

53. In what follows, LCA is the category of locally compact abelian groups and continuous group homomorphisms.

54. Abelian Categories.

55. Hence each non-Abelian finite simple group has order divisible by at least three distinct primes.

56. The dual group of a locally compact abelian group is used as the underlying space for an abstract version of the Fourier transform.

57. Both the Abelian and non-Abelian cases are considered.

58. Generally speaking, negation is an automorphism of any abelian group, but not of a ring or field.

59. Its recurrent configurations form an Abelian group, whose identity is a fractal composed of self-similar patches.

60. BS(1, 1) is the free abelian group on two generators, and BS(1, −1) is the fundamental group of the Klein bottle.

61. In general a group G is free Abelian if G ∼= F ab (S) for some set S

62. A divisor D is an element of the free abelian group on the points of the surface.

63. Thus, we obtain the global reciprocity map of the idele class group to the abelian part of the absolute Galois group of the field.

64. An example of a non-abelian, pre-abelian category is, once again, the category of topological abelian groups.

65. Class field theory is a branch of algebraic number theory which seeks to classify all the abelian extensions of a given algebraic number field, meaning Galois extensions with abelian Galois group.

66. We treat gauge field in two cases:a) Abelian andb) non-Abelian.

67. It is an open problem whether every non-abelian p-group G has an automorphism of order pp.

68. Expressed algebraically, that same equation would be:

69. For general non-abelian locally compact groups, harmonic analysis is closely related to the theory of unitary group representations.

70. The category of all finitely generated abelian groups is also an abelian category, as is the category of all finite abelian groups.

71. Abelian (Vice-President)

72. Any abelian variety is isogenous to a product of simple abelian varieties.

73. I give two such notions, abelian and 2-abelian groupoid enriched categories.

74. Thus, an Abelian variety can be imbedded as a closed subvariety in a projective space; each rational mapping of a non-singular variety into an Abelian variety is regular; the group law on

75. The Witt group of k is the abelian group W(k) of equivalence classes of non-degenerate symmetric bilinear forms, with the group operation corresponding to the orthogonal direct sum of forms.

76. A finite field F is not algebraically closed.

77. If a countable discrete group contains a (non-abelian) free subgroup on two generators, then it is not amenable.

78. In mathematics, a Witt group of a field, named after Ernst Witt, is an abelian group whose elements are represented by symmetric bilinear forms over the field.

79. Abelian is a Python library for computations on elementary locally compact Abelian groups (LCAs)

80. H.E. Mr. Movses Abelian Ambassador Extraordinary and Plenipotentiary Permanent Representative ( # anuary # ) Mrs. Rouzanna Abelian