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.
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.
The relation »is subquotient of« is transitive.
Let groups and and be group homomorphisms, then also the composition
is a homomorphism.
If is a subgroup of and a subgroup of , then is a subgroup of . We have , indeed , because every has a preimage in . Thus . This means that the image, say , of a subgroup, say , of is also the image of a subgroup, namely under , of .
In other words: If is a subquotient of and is subquotient of then is subquotient of . ■
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