Definition
Let N be a normal subgroup of a group G. We define the set G/N to be the set of all left cosets of N in G, i.e., G/N = { aN : a in G }. The group operation on G/N is the product of subsets defined above. In other words, for each aN and bN in G/N, the product of aN and bN is (aN)(bN). This operation is closed, because (aN)(bN) really is a left coset:
- (aN)(bN) = a(Nb)N = a(bN)N = (ab)NN = (ab)N.
The normality of N is used in this equation. Because of the normality of N, the left cosets and right cosets of N in G are equal, and so G/N could be defined as the set of right cosets of N in G. Because the operation is derived from the product of subsets of G, the operation is well-defined (does not depend on the particular choice of representatives), associative, and has identity element N. The inverse of an element aN of G/N is a−1N.
For example, consider the group with addition modulo 6:
- G = {0, 1, 2, 3, 4, 5}.
Let
- N = {0, 3}.
The quotient group is:
- G/N = { aN : a ∈ G } = { a{0, 3} : a ∈ {0, 1, 2, 3, 4, 5} } =
- { 0{0, 3}, 1{0, 3}, 2{0, 3}, 3{0, 3}, 4{0, 3}, 5{0, 3} } =
- { {(0+0) mod 6, (0+3) mod 6}, {(1+0) mod 6, (1+3) mod 6},
- {(2+0) mod 6, (2+3) mod 6}, {(3+0) mod 6, (3+3) mod 6},
- {(4+0) mod 6, (4+3) mod 6}, {(5+0) mod 6, (5+3) mod 6} } =
- { {0, 3}, {1, 4}, {2, 5}, {3, 0}, {4, 1}, {5, 2} } =
- { {0, 3}, {1, 4}, {2, 5}, {0, 3}, {1, 4}, {2, 5} } =
- { {0, 3}, {1, 4}, {2, 5} }.
The basic argument above is still valid if G/N is defined to be the set of all right cosets.
Read more about this topic: Quotient Group
Famous quotes containing the word definition:
“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)
“Scientific method is the way to truth, but it affords, even in
principle, no unique definition of truth. Any so-called pragmatic
definition of truth is doomed to failure equally.”
—Willard Van Orman Quine (b. 1908)
“Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
—The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)