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### Is every abelian group is cyclic or not and why?

every abelian group is not cyclic. e.g, set of (Q,+) it is an abelian group but not cyclic. Full Answer

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Abelian meaning commutative. If the symmetry group of a square is commutative then it's an abelian group or else it's not. Full Answer

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An abelian group is a group in which ab = ba for all members a and b of the group. Full Answer

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a group or set of group is said to be abelian if the law of commutation is always held. it means if for a group or set S having elements a,b belongs to S then a*b=b*a then the group is… Full Answer

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The abelian groups of order 24 are C3xC8, C2xC12, C2xC2xC6. There are other 12 non-abelian groups of order 24 Full Answer

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No. Full Answer

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No. Full Answer

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No, for instance the Klein group is finite and abelian but not cyclic. Even more groups can be found having this chariacteristic for instance Z9 x Z9 is abelian but not cyclic Full Answer

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An abelianization is a homomorphism which transforms a group into an abelian group. Full Answer

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Abelianization is a homomorphism which transforms a group into an Abelian group. Full Answer

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The term abelian is most commonly encountered in group theory, where it refers to a specific type of group known as an abelian group. An abelian group, simply put, is a commutative group, meaning that when the group operation is… Full Answer

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The set of integers, under addition. Full Answer

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By LaGrange's Thm., the order of an element of a group must divide the order of the group. Since 3 is prime, up to isomorphism, the only group of order three is {1,x,x^2} where x^3=1. Note that this is a… Full Answer

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abelian group Full Answer

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An additive group is an abelian group when it is written using the + symbol for its binary operation. Full Answer

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prove a group of order9 is abelian Full Answer

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There are 5 groups of order 8 up to isomorphism. 3 abelian ones (C8, C4xC2, C2xC2xC2) and 2 non-abelian ones (dihedral group D8 and quaternion group Q) Full Answer

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Algebra

If you are working with real numbers, or even complex numbers, pq is the same as qp, so the result is the same as 2pq. If you use some multiplication that is NOT commutative (such as, when you multiply matrices)… Full Answer

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A bimodule is an abelian group which is both a left and right module, such as the left and right multiplications are compatible. Full Answer

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Hovhannes Abelian was born in 1865. Full Answer

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Hovhannes Abelian died in 1936. Full Answer

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Definitions

Abelian algebra is a form of algebra in which the multiplication within an expression is commutative. Full Answer

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Joel Samuel Georges has written: 'Associativity conditions for division algebras corresponding to any Abelian group' -- subject(s): Abelian groups, Universal Algebra 'Introductory mathematical analysis' -- subject(s): Calculus, Functions Full Answer

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Paul Edwin Lewis has written: 'Characters of Abelian groups' -- subject(s): Group theory Full Answer

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An Abelian is a member of a fourth-century African sect mentioned by St. Augsustine, who married but lived in continence. Full Answer

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The Abelian property, after the Norwegian mathematician, Niels Abel (early 19th Century) who made major contributions to Group Theory. Full Answer

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A. P. Mishina has written: 'Higher algebra' 'Abelian groups and modules' -- subject(s): Abelian groups, Modules (Algebra) Full Answer

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A. B. Katok has written: 'Lectures on surfaces' -- subject(s): Surfaces 'Rigidity in higher rank Abelian group actions' -- subject(s): Rigidity (Geometry), Abelian groups 'Invariant manifolds, entropy, and billiards' -- subject(s): Entropy, Global analysis (Mathematics), Invariant manifolds, Ergodic theory, Differentiable… Full Answer

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H. Inassaridze has written: 'Non-Abelian homological algebra and its applications' -- subject(s): Algebra, Homological, Homological Algebra, Non-Abelian groups Full Answer

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Francis Borceux has written: 'Mal'cev, protomodular, homological and semi-abelian categories' -- subject(s): Abelian categories, Categories (Mathematics), Homological Algebra Full Answer

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Yes. Lets call the generator of the group z, then every element of the group can be written as zk for some k. Then the product of two elements is: zkzm=zk+m Notice though that then zmzk=zm+k=zk+m=zkzm, so the group is… Full Answer

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More precisely, I think you're asking whether the set of n X n matrices forms an abelian group under multiplication. The answer is no (assuming n>1). For example (1 0)(0 1) = (0 1) (0 0)(0 0) (0 0), but… Full Answer

Abelian (meaning commutative: a + b = b + a). Full Answer

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A non-Abelian grape. (It doesn't commute.) Full Answer

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Christina Birkenhake has written: 'Complex Abelian varieties' -- subject(s): Abelian varieties, Riemann surfaces 'Complex tori' -- subject(s): Complex manifolds, Torus (Geometry) Full Answer

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Henry Frederick Baker has written: 'Abel's theorem and the allied theory' -- subject(s): Abelian Functions, Functions, Abelian, Functions, Theta, Riemann surfaces, Theta Functions Full Answer

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The Abelian or commutative property of multiplication of numbers. Full Answer

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The generalisation is that multiplication is Abelian (or commutative) for numbers. Full Answer

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Bernard Kin-sion Cheung has written: 'On the structure of multiplicative cohomology theory with abelian coefficients' -- subject(s): Abelian groups, Homology theory, Homotopy theory Full Answer

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In
Definitions

Abelianness is the quality of an expression being abelian - having a commutative-defining operation. Full Answer

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An Abelonian is an alternative term for an Abelian - a member of a historical Roman Catholic sect. Full Answer

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The Abelian property or commutative property. Full Answer

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Commutative property or Abelian property. Full Answer

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Commutativity or Abelian property Full Answer

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The Abelian property or commutativity. Full Answer