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Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras.The phrase abstract algebra was coined at the turn of the 20th century to distinguish this area from what was normally referred to as algebra, the study of the rules for manipulating formulae and algebraic expressions involving unknowns and ...
Among mathematical objects are numbers, functions, sets, a great variety of things called "spaces" of one kind or another, algebraic structures such as rings, groups, or fields, and many other things. List of mathematical examples; List of algebraic surfaces; List of curves; List of complex reflection groups; List of complexity classes
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics ...
In philosophy and the arts, a fundamental distinction is between things that are abstract and things that are concrete.While there is no general consensus as to how to precisely define the two, examples include that things like numbers, sets, and ideas are abstract objects, while plants, dogs, and planets are concrete objects. [1]
Abstraction is an ongoing process in mathematics and the historical development of many mathematical topics exhibits a progression from the concrete to the abstract. For example, the first steps in the abstraction of geometry were historically made by the ancient Greeks, with Euclid's Elements being the earliest extant documentation of the ...
Convolution. Cauchy product –is the discrete convolution of two sequences; Farey sequence – the sequence of completely reduced fractions between 0 and 1; Oscillation – is the behaviour of a sequence of real numbers or a real-valued function, which does not converge, but also does not diverge to +∞ or −∞; and is also a quantitative measure for that.
In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods ...
Once this abstract group concept emerged, results were reformulated in this abstract setting. For example, Sylow's theorem was reproven by Frobenius in 1887 directly from the laws of a finite group, although Frobenius remarked that the theorem followed from Cauchy's theorem on permutation groups and the fact that every finite group is a ...