Product of Subsets of A Group
In the following discussion, we will use a binary operation on the subsets of G: if two subsets S and T of G are given, we define their product as ST = {st : s ∈ S ∧ t ∈ T}. This operation is associative and has as identity element the singleton {e}, where e is the identity element of G. Thus, the set of all subsets of G forms a monoid under this operation.
In terms of this operation we can first explain what a quotient group is, and then explain what a normal subgroup is:
- A quotient group of a group G is a partition of G which is itself a group under this operation.
It is fully determined by the subset containing e. A normal subgroup of G is the set containing e in any such partition. The subsets in the partition are the cosets of this normal subgroup.
A subgroup N of a group G is normal if and only if the coset equality aN = Na holds for all a in G. In terms of the binary operation on subsets defined above, a normal subgroup of G is a subgroup that commutes with every subset of G and is denoted N ◁ G. A subgroup that permutes with every subgroup of G is called a permutable subgroup.
Read more about this topic: Quotient Group
Famous quotes containing the words product of, product and/or group:
“The UN is not just a product of do-gooders. It is harshly real. The day will come when men will see the UN and what it means clearly. Everything will be all right—you know when? When people, just people, stop thinking of the United Nations as a weird Picasso abstraction, and see it as a drawing they made themselves.”
—Dag Hammarskjöld (1905–1961)
“Humour is the describing the ludicrous as it is in itself; wit is the exposing it, by comparing or contrasting it with something else. Humour is, as it were, the growth of nature and accident; wit is the product of art and fancy.”
—William Hazlitt (1778–1830)
“He hung out of the window a long while looking up and down the street. The world’s second metropolis. In the brick houses and the dingy lamplight and the voices of a group of boys kidding and quarreling on the steps of a house opposite, in the regular firm tread of a policeman, he felt a marching like soldiers, like a sidewheeler going up the Hudson under the Palisades, like an election parade, through long streets towards something tall white full of colonnades and stately. Metropolis.”
—John Dos Passos (1896–1970)