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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 .
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]
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 ...
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.
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.
In abstract algebra, the fundamental theorem on homomorphisms, also known as the fundamental homomorphism theorem, or the first isomorphism theorem, relates the structure of two objects between which a homomorphism is given, and of the kernel and image of the homomorphism. The homomorphism theorem is used to prove the isomorphism theorems.
The direct sum is an operation between structures in abstract algebra, a branch of mathematics.It is defined differently but analogously for different kinds of structures. As an example, the direct sum of two abelian groups and is another abelian group consisting of the ordered pairs (,) where and
G is the group /, the integers mod 8 under addition. The subgroup H contains only 0 and 4, and is isomorphic to /.There are four left cosets of H: H itself, 1+H, 2+H, and 3+H (written using additive notation since this is an additive group).