Subquotient

In the mathematical sub-areas of category theory and abstract algebra , a subquotient is understood to be a quotient object of a sub-object.

In the language of group theory , a subobject is a subgroup and a quotient object is a quotient group (also called a factor group ): Thus, a subquotient of a group is the image of a subgroup of under a group homomorphism . ${\displaystyle G}$ ${\displaystyle G}$

The term subquotient is used u. a. in the classification of the finite simple groups , especially in the sporadic groups .

definition

Group theory

Is a group, a subgroup of, and a normal subgroup of , in characters ${\displaystyle G}$ ${\displaystyle G'}$ ${\displaystyle G}$ ${\displaystyle G''}$ ${\displaystyle G'}$

${\displaystyle G\geq G'\vartriangleright G'',}$

then the factor group (quotient group) is called a subquotient of . ${\displaystyle H:=G'/G''}$ ${\displaystyle G}$

In the literature on sporadic groups there are formulations such as

1. ${\displaystyle G}$ involved ${\displaystyle H}$
2. ${\displaystyle H}$ is involved in ${\displaystyle G}$

for the same fact.

Module theory

Be a ring with one element . The modules have - sub- modules and - quotient modules (factor modules ). The subquotients are defined in the same way as for the groups . ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle R}$

The formation of the term also applies to a non-commutative ring and left / right-side modules above this ring.

Properties and examples

• A sub-object of as well as a (homomorphic) image of is a subquotient of ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle G.}$

• The factors of a subnormal series are subquotients.

• The butterfly lemma makes a statement about the isomorphism of certain subquotients.

Finite objects

If all objects have finite cardinalities , then there are formulas that relate them to indices , see for example Lagrange's theorem . Because of the above designations ${\displaystyle |H|=[H:1]=[G':G'']}$

${\displaystyle |G|=[G:1]=[G:G']\cdot [G':G'']\cdot [G'':1]=[G:G']\cdot |H|\cdot |G''|,}$

and is in particular a divisor of as well ${\displaystyle |H|}$ ${\displaystyle |G|}$ ${\displaystyle |H|\leq |G|.}$

Partial order

For finite objects the relation "is subquotient of" is an order relation, namely a partial order .

Reflexivity

${\displaystyle G/\{1\}}$is subquotient of . ${\displaystyle G}$

Transitivity

Sub-quotients of sub-quotients are sub-quotients.

Proof for groups

Let be the subquotient of and the canonical homomorphism. Is now , that is, a subquotient of , then is first ${\displaystyle H:=G'/G''}$ ${\displaystyle G}$ ${\displaystyle \phi \colon G'\to H}$ ${\displaystyle H\geq H'\vartriangleright H''}$ ${\displaystyle H'/H''}$ ${\displaystyle H}$

${\displaystyle {\begin{array}{ccccccccc}G\quad &\geq &\quad \;G'\quad &\geq &\phi ^{-1}(H')&\geq &\phi ^{-1}(H'')\;\;&\vartriangleright \quad &G''\\&\phi \!:&{\Big \downarrow }&&{\Big \downarrow }&&{\Big \downarrow }&&{\Big \downarrow }\\&&H&\geq &H'&\vartriangleright &H''&\vartriangleright \quad &\{1\}\\\end{array}}}$

with each vertical ( ) figure with matching surjective for the couple . ${\displaystyle \downarrow }$ ${\displaystyle \phi \!:X\to Y,g\mapsto g\,G''}$ ${\displaystyle g\in G'}$ ${\displaystyle (X,Y)={\bigl (}G',H{\bigr )},{\bigl (}\phi ^{-1}(H'),H'{\bigr )},{\bigl (}\phi ^{-1}(H''),H''{\bigr )},{\bigl (}G'',\{1\}{\bigr )}}$

Now are the archetypes and subgroups of that contain. Furthermore, there is and , since all have a prototype in . Furthermore, is a normal divisor of . So the subquotient of is considered a subquotient of . ${\displaystyle \phi ^{-1}(H')}$ ${\displaystyle \phi ^{-1}(H'')}$ ${\displaystyle G'}$ ${\displaystyle G''}$ ${\displaystyle \phi (\phi ^{-1}(H'))=H'}$ ${\displaystyle \phi (\phi ^{-1}(H''))=H''}$ ${\displaystyle h\in H}$ ${\displaystyle G'}$ ${\displaystyle \phi ^{-1}(H'')}$ ${\displaystyle \phi ^{-1}(H')}$ ${\displaystyle H'/H''}$ ${\displaystyle H}$ ${\displaystyle H'/H''\cong \phi ^{-1}(H')/\phi ^{-1}(H'')}$ ${\displaystyle G}$

Antisymmetry

If two objects are subquotients of each other, they are isomorphic .

proof

The correlation between and can only be maintained because of , that is , with and , from which it follows. ${\displaystyle G}$ ${\displaystyle H}$ ${\displaystyle |H|\leq |G|\leq |H|}$ ${\displaystyle |H|=|G|}$ ${\displaystyle [G:G']=1,[G'':1]=1}$ ${\displaystyle [H:H']=1,[H'':1]=1}$ ${\displaystyle G\cong H}$

Discrete order

The order relation "is subquotient of" is a discrete order in finite groups ; H. the order topology it creates is a discrete topology . In formulas and with and as relation symbols: ${\displaystyle \preceq }$ ${\displaystyle \prec }$

If there is one with such a way that ${\displaystyle I\prec G,}$ ${\displaystyle H\prec G}$ ${\displaystyle I\preceq H}$ ${\displaystyle H\preceq H'\prec G\implies H'=H.}$

Such is called a maximum real subquotient of . The term is needed , for example, when arranging the sporadic groups in the Hasse diagram . ${\displaystyle H}$ ${\displaystyle G}$

Individual evidence

1. ↑ Dieter Held: The Classification of Finite Simple Groups (PDF, 131 kB) p. 19 ( Memento of the original from June 26, 2013 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. 
2. ^ Robert Griess: The Friendly Giant . In: Inventiones Mathematicae . tape 69 , 1982, pp. 91 , doi : 10.1007 / BF01389186 ( online at digizeitschriften.de ).
3. ↑ Noether's Isomorphy Theorems