FACTOR GROUP Meaning and
Definition
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A factor group, also known as a quotient group, is a fundamental concept in group theory. It is a mathematical structure that arises from dividing a group into subsets based on the equivalence of certain elements.
Formally, let G be a group with an operation * and H be a subgroup of G. The factor group of G by H, denoted by G/H, is the set of all cosets of H in G, with the operation of coset multiplication defined as follows: for any two cosets aH and bH in G/H, their product (aH)(bH) is defined as (ab)H. This operation is well-defined, meaning that the result does not depend on the choice of representatives a and b.
The factor group G/H inherits certain properties from the original group G. It forms a group itself, with the identity element being the coset containing the identity of G. The factor group operation is associative, and each coset has an inverse. Furthermore, the factor group exhibits closure, meaning that the multiplication of two cosets is always another coset.
Factor groups play a crucial role in group theory and allow for the analysis and classification of groups. They help identify the similarities and differences between different groups and provide a convenient way to study their structure. By examining the factor groups of a given group, mathematicians can gain insights into its symmetries, subgroups, and other important group-theoretic properties.
Common Misspellings for FACTOR GROUP
- dactor group
- cactor group
- vactor group
- gactor group
- tactor group
- ractor group
- fzctor group
- fsctor group
- fwctor group
- fqctor group
- faxtor group
- favtor group
- faftor group
- fadtor group
- facror group
- facfor group
- facgor group
- facyor group
- fac6or group
- fac5or group
Etymology of FACTOR GROUP
The term "factor group" is a mathematical concept derived from the word "factor" and "group".
The word "factor" comes from the Latin word "factor", which means "maker" or "doer". In mathematics, a factor refers to a number or quantity that divides another number or quantity evenly. For example, in the equation 6 = 2 x 3, 2 and 3 are the factors of 6.
The word "group" has its roots in the Middle English word "group", which came from the French word "groupe", and ultimately from the Italian word "gruppo", meaning "cluster" or "bunch". In mathematics, a group is a set of elements combined with a binary operation (such as addition or multiplication) that satisfies certain conditions, such as closure, associativity, identity, and invertibility.
