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. So in the algebraic structure of groups, H {\displaystyle H} is a subquotient of G {\displaystyle G} if there exists a subgroup G ′ {\displaystyle G'} of G {\displaystyle G} and a normal subgroup G ″ {\displaystyle G''} of G ′ {\displaystyle G'} so that H {\displaystyle H} is isomorphic to G ′ / G ″ {\displaystyle G'/G''} . In the literature about sporadic groups wordings like " H {\displaystyle H} is involved in G {\displaystyle G} " can be found with the apparent meaning of " H {\displaystyle H} is a subquotient of G {\displaystyle G} ". As in the context of subgroups, in the context of subquotients the term trivial may be used for the two subquotients G {\displaystyle G} and { 1 } {\displaystyle \{1\}} which are present in every group G {\displaystyle G} . 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.
Example There are subquotients of groups which are neither a subgroup nor a quotient of it. For example, according to the article Sporadic group, Fi22 has a double cover which is a subgroup of Fi23, so it is a subquotient of Fi23 without being a subgroup or quotient of it.
Order relation The relation subquotient of is an order relation, which shall be denoted by ⪯ {\displaystyle \preceq } . It shall be proved for groups.
Notation Let G be a group, let G′ be a subgroup of G, let G′′ be a normal subgroup of G′, and let H be the quotient group G′ / G′′. Then we say that H is a subquotient of G. In symbols, let G′′ ◃ G′ ≤ G and H = G′ / G′′; then H ⪯ G. This relationship has the following properties: Reflexivity: G ⪯ G {\displaystyle G\preceq G} , i. e. every element is related to itself. Indeed, G {\displaystyle G} is isomorphic to the subquotient G / { 1 } {\displaystyle G/\{1\}} of G {\displaystyle G} . Antisymmetry: if G ⪯ H {\displaystyle G\preceq H} and H ⪯ G {\displaystyle H\preceq G} then G ≅ H {\displaystyle G\cong H} ; that is, no two distinct elements precede each other. Indeed, a comparison of the group orders of G {\displaystyle G} and H {\displaystyle H} then yields | G | = | H | {\displaystyle |G|=|H|} from which G ≅ H {\displaystyle G\cong H} . Transitivity: if H ′ / H ″ ⪯ H {\displaystyle H'/H''\preceq H} and H ⪯ G {\displaystyle H\preceq G} then H ′ / H ″ ⪯ G {\displaystyle H'/H''\preceq G} .
Proof of transitivity for groups Let H ′ / H ″ {\displaystyle H'/H''} be a subquotient of H {\displaystyle H} , let H := G ′ / G ″ {\displaystyle H:=G'/G''} be a subquotient of G {\displaystyle G} , and let φ : G ′ → H {\displaystyle \varphi \colon G'\to H} be the canonical homomorphism. Then in the following diagram, all vertical ( ↓ {\displaystyle \downarrow } ) maps φ : X → Y , x ↦ x G ″ {\displaystyle \varphi \colon X\to Y,\;x\mapsto x\,G''}
are surjective for the respective pairs
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