Subquotient

In the <a href="/facts/Mathematics/pxTouaz4">mathematical</a> fields of <a href="/facts/Category_theory/6zS3DXPa">category theory</a> and <a href="/facts/Abstract_algebra/YNNE1CNz">abstract algebra</a>, a subquotient is a <a href="/facts/Quotient_object/lEcCIhrP">quotient object</a> of a <a href="/facts/Subobject/lEcCIhrP">subobject</a>. Subquotients are particularly important in <a href="/facts/Abelian_categories/4WN473Yc">abelian categories</a>, and in <a href="/facts/Group_theory/NfowcI5H">group theory</a>, where they are also known as sections, though this conflicts with <a href="/facts/Section_(category_theory)/Ee1wexPw">a different meaning</a> in category theory.
So in the <a href="/facts/Algebraic_structure/TkuXca24">algebraic structure</a> of groups, 
 
 
 
 H
 
 
 {\displaystyle H}
 
 is a subquotient of 
 
 
 
 G
 
 
 {\displaystyle G}
 
 if there exists a <a href="/facts/Subgroup/r91mBgpa">subgroup</a> 
 
 
 
 
 G
 ′
 
 
 
 {\displaystyle G'}
 
 of 
 
 
 
 G
 
 
 {\displaystyle G}
 
 and a <a href="/facts/Normal_subgroup/aRDaGwLm">normal subgroup</a> 
 
 
 
 
 G
 ″
 
 
 
 {\displaystyle G''}
 
 of 
 
 
 
 
 G
 ′
 
 
 
 {\displaystyle G'}
 
 so that 
 
 
 
 H
 
 
 {\displaystyle H}
 
 is <a href="/facts/Isomorphic/pj8KgkxU">isomorphic</a> to 
 
 
 
 
 G
 ′
 
 
 /
 
 
 G
 ″
 
 
 
 {\displaystyle G'/G''}
 
.
In the literature about <a href="/facts/Sporadic_groups/hjNHTKVv">sporadic groups</a> 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., <a href="/facts/Harish-Chandra/f9De1zCJ">Harish-Chandra</a>'s subquotient theorem.

Subquotient open-in-new

Subquotient