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In abstract algebra, every subgroup of a cyclic group is cyclic. Moreover, for a finite cyclic group of order n, every subgroup's order is a divisor of n, and there is exactly one subgroup for each divisor. [1] [2] This result has been called the fundamental theorem of cyclic groups. [3] [4]
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations acting on their elements. [1] Algebraic structures include groups , rings , fields , modules , vector spaces , lattices , and algebras over a field .
An abstract group defined by this multiplication is often denoted C n, and we say that G is isomorphic to the standard cyclic group C n. Such a group is also isomorphic to Z / n Z , the group of integers modulo n with the addition operation, which is the standard cyclic group in additive notation.
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 ...
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) [1] is a subgroup that is invariant under conjugation by members of the group of which it is a part.
Algebra includes the study of algebraic structures, which are sets and operations defined on these sets satisfying certain axioms. The field of algebra is further divided according to which structure is studied; for instance, group theory concerns an algebraic structure called group. Outline of algebra; Glossary of field theory; Glossary of ...
Contemporary Abstract Algebra (6e ed.). Houghton Mifflin. ISBN 0-618-51471-6. Linear algebra theory. Explains commutativity in chapter 1, uses it throughout. Goodman, Frederick (2003). Algebra: Abstract and Concrete, Stressing Symmetry (2e ed.). Prentice Hall. ISBN 0-13-067342-0. Abstract algebra theory. Uses commutativity property throughout book.
Unlike the articles by Study and Cartan, which were exploratory, Brauer’s article reads as a modern abstract algebra text with its universal coverage. Consider his introduction: In the beginning of the 19th century, the usual complex numbers and their introduction through computations with number-pairs or points in the plane, became a general ...