subquotient について

Words near each other

Dictionary Lists

mini英和辞書

翻訳と辞書　辞書検索 [ 開発暫定版 ]

スポンサードリンク

subquotient ：ウィキペディア英語版

subquotient
In the mathematical fields of category theory and abstract algebra, a subquotient is a quotient object of a subobject. Subquotients are particularly important in abelian categories, and in group theory, where they are also known as sections, though this conflicts with a different meaning in category theory.
For example, of the 26 sporadic groups, 20 are subquotients of the monster group, and are referred to as the "Happy Family", while the other 6 are pariah groups.
A quotient of a subrepresentation of a representation (of, say, a group) might be called a subquotient representation; e.g., Harish-Chandra's subquotient theorem.
==Transitive relation==
The relation »is subquotient of« is transitive.
;Proof
Let

G,H,J

groups and

\phi\colon G\to H

and

\psi\colon H\to J

be group homomorphisms, then also the composition
:

\psi\circ\phi\colon G\to J, g \mapsto (\psi\circ\phi)(g):=\psi(\phi(g))

is a homomorphism.
If

U

is a subgroup of

G

and

V

a subgroup of

\phi(U)

, then

U':=\phi^(V)

is a subgroup of

U\leq G

. We have

\phi(U')\subseteq V

, indeed

\phi(U')=V

, because every

v \in V\subseteq \phi(U)

has a preimage in

U

. Thus

(\psi\circ\phi)(U')=\psi(V)

. This means that the image, say

\psi(V)

, of a subgroup, say

V

, of

H

is also the image of a subgroup, namely

U'

under

\psi\circ\phi

, of

G

.
In other words: If

\psi(V)

is a subquotient of

\phi(U)

and

\phi(U)

is subquotient of

G

then

\psi(V)

is subquotient of

G

. ■

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
■ウィキペディアで「subquotient」の詳細全文を読む

スポンサードリンク

翻訳と辞書 : 翻訳のためのインターネットリソース