QUOTIENT GROUP Meaning and
Definition
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A quotient group, also known as a factor group, is a fundamental concept in abstract algebra, specifically in group theory. It is formed by partitioning a group and then performing group operations on the resulting partitions.
More precisely, let G be a group and H be a normal subgroup of G. The quotient group, denoted as G/H, is the group formed by partitioning G into distinct cosets of H and defining a binary operation on the set of these cosets. Each coset is formed by multiplying the elements of H by a fixed element of G. The binary operation on the cosets is defined by multiplying their respective representative elements in G.
The defining property of a quotient group is that the operation is well-defined and it inherits closure, associativity, and inverses from the original group G. The identity element of the quotient group is the coset that contains the identity element of G. As H is a normal subgroup, the quotient group G/H is also a group.
Quotient groups are most often used to study the relationships between different groups, as they allow the reduction of complex groups into simpler ones. Understanding the structure of quotient groups contributes to the understanding of the original group's properties and its subgroups. The significance of quotient groups extends into various areas of mathematics, including number theory, algebraic geometry, and algebraic topology.
Common Misspellings for QUOTIENT GROUP
- 1uotient group
- 2uotient group
- wuotient group
- auotient group
- qyotient group
- qhotient group
- qjotient group
- qiotient group
- q8otient group
- q7otient group
- quitient group
- quktient group
- qultient group
- quptient group
- qu0tient group
- qu9tient group
- quorient group
- quofient group
- quogient group
- quoyient group
Etymology of QUOTIENT GROUP
The word "quotient group" is a term used in abstract algebra to describe a specific type of mathematical construction. Its etymology lies in two key aspects: "quotient" and "group".
- "Quotient": The term "quotient" refers to the result of dividing one number or quantity by another. In mathematics, it is used to denote the result of a division operation. The word "quotient" itself comes from the Latin word "quotiens", meaning "how many times" or "how often".
- "Group": In mathematics, a group is an algebraic structure consisting of a set with an operation that satisfies certain properties, such as closure, associativity, identity element, and invertibility. The concept of a group originates from the Latin word "gruppus", which referred to a knot or a cluster.
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