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}
“[ 1] 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}
.[citation needed ]
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.[ 2]
There are subquotients of groups which are neither subgroup nor quotient of it. E. g. according to article Sporadic group , Fi 22 has a double cover which is a subgroup of Fi 23 , so it is a subquotient of Fi 23 without being a subgroup or quotient of it.
The relation subquotient of is an order relation – which shall be denoted by
⪯
{\displaystyle \preceq }
. It shall be proved for groups.
Notation
For group
G
{\displaystyle G}
, subgroup
G
′
{\displaystyle G'}
of
G
{\displaystyle G}
(
⇔:
G
′
≤
G
)
{\displaystyle (\Leftrightarrow :G'\leq G)}
and normal subgroup
G
″
{\displaystyle G''}
of
G
′
{\displaystyle G'}
(
⇔:
G
″
⊲
G
′
)
{\displaystyle (\Leftrightarrow :G''\vartriangleleft G')}
the quotient group
H
:=
G
′
/
G
″
{\displaystyle H:=G'/G''}
is a subquotient of
G
{\displaystyle G}
, i. e.
H
⪯
G
{\displaystyle H\preceq G}
.
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}
, i. e. 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 [ edit ]
Let
H
′
/
H
″
{\displaystyle H'/H''}
be subquotient of
H
{\displaystyle H}
, furthermore
H
:=
G
′
/
G
″
{\displaystyle H:=G'/G''}
be subquotient of
G
{\displaystyle G}
and
φ
:
G
′
→
H
{\displaystyle \varphi \colon G'\to H}
be the canonical homomorphism . Then all vertical (
↓
{\displaystyle \downarrow }
) maps
φ
:
X
→
Y
,
x
↦
x
G
″
{\displaystyle \varphi \colon X\to Y,\;x\mapsto x\,G''}
G
″
{\displaystyle G''}
≤
{\displaystyle \leq }
φ
−
1
(
H
″
)
{\displaystyle \varphi ^{-1}(H'')}
≤
{\displaystyle \leq }
φ
−
1
(
H
′
)
{\displaystyle \varphi ^{-1}(H')}
⊲
{\displaystyle \vartriangleleft }
G
′
{\displaystyle G'}
φ
:
{\displaystyle \varphi \!:}
↓
{\displaystyle {\Big \downarrow }}
↓
{\displaystyle {\Big \downarrow }}
↓
{\displaystyle {\Big \downarrow }}
↓
{\displaystyle {\Big \downarrow }}
{
1
}
{\displaystyle \{1\}}
≤
{\displaystyle \leq }
H
″
{\displaystyle H''}
⊲
{\displaystyle \vartriangleleft }
H
′
{\displaystyle H'}
⊲
{\displaystyle \vartriangleleft }
H
{\displaystyle H}
are surjective for the respective pairs
(
X
,
Y
)
∈
{\displaystyle (X,Y)\;\;\;\in }
{
(
G
″
,
{
1
}
)
{\displaystyle {\Bigl \{}{\bigl (}G'',\{1\}{\bigr )}{\Bigr .}}
,
{\displaystyle ,}
(
φ
−
1
(
H
″
)
,
H
″
)
{\displaystyle {\bigl (}\varphi ^{-1}(H''),H''{\bigr )}}
,
{\displaystyle ,}
(
φ
−
1
(
H
′
)
,
H
′
)
{\displaystyle {\bigl (}\varphi ^{-1}(H'),H'{\bigr )}}
,
{\displaystyle ,}
(
G
′
,
H
)
}
.
{\displaystyle {\Bigl .}{\bigl (}G',H{\bigr )}{\Bigr \}}.}
The preimages
φ
−
1
(
H
′
)
{\displaystyle \varphi ^{-1}\left(H'\right)}
and
φ
−
1
(
H
″
)
{\displaystyle \varphi ^{-1}\left(H''\right)}
are both subgroups of
G
′
{\displaystyle G'}
containing
G
″
,
{\displaystyle G'',}
and it is
φ
(
φ
−
1
(
H
′
)
)
=
H
′
{\displaystyle \varphi \left(\varphi ^{-1}\left(H'\right)\right)=H'}
and
φ
(
φ
−
1
(
H
″
)
)
=
H
″
,
{\displaystyle \varphi \left(\varphi ^{-1}\left(H''\right)\right)=H'',}
because every
h
∈
H
{\displaystyle h\in H}
has a preimage
g
∈
G
′
{\displaystyle g\in G'}
with
φ
(
g
)
=
h
.
{\displaystyle \varphi (g)=h.}
Moreover, the subgroup
φ
−
1
(
H
″
)
{\displaystyle \varphi ^{-1}\left(H''\right)}
is normal in
φ
−
1
(
H
′
)
.
{\displaystyle \varphi ^{-1}\left(H'\right).}
As a consequence, the subquotient
H
′
/
H
″
{\displaystyle H'/H''}
of
H
{\displaystyle H}
is a subquotient of
G
{\displaystyle G}
in the form
H
′
/
H
″
≅
φ
−
1
(
H
′
)
/
φ
−
1
(
H
″
)
.
{\displaystyle H'/H''\cong \varphi ^{-1}\left(H'\right)/\varphi ^{-1}\left(H''\right).}
Relation to cardinal order [ edit ]
In constructive set theory , where the law of excluded middle does not necessarily hold, one can consider the relation subquotient of as replacing the usual order relation (s) on cardinals . When one has the law of the excluded middle, then a subquotient
Y
{\displaystyle Y}
of
X
{\displaystyle X}
is either the empty set or there is an onto function
X
→
Y
{\displaystyle X\to Y}
. This order relation is traditionally denoted
≤
∗
.
{\displaystyle \leq ^{\ast }.}
If additionally the axiom of choice holds, then
Y
{\displaystyle Y}
has a one-to-one function to
X
{\displaystyle X}
and this order relation is the usual
≤
{\displaystyle \leq }
on corresponding cardinals.
^ Griess, Robert L. (1982), "The Friendly Giant" , Inventiones Mathematicae , 69 : 1−102, Bibcode :1982InMat..69....1G , doi :10.1007/BF01389186 , hdl :2027.42/46608 , S2CID 123597150
^ Dixmier, Jacques (1996) [1974], Enveloping algebras , Graduate Studies in Mathematics , vol. 11, Providence, R.I.: American Mathematical Society , ISBN 978-0-8218-0560-2 , MR 0498740 p. 310