From Wikipedia, the free encyclopedia
Jump to: navigation, search

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[edit]

The relation »is subquotient of« is transitive.


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^{-1}(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.  ■

See also[edit]