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,
is a subquotient of
if there exists a subgroup
of
and a normal subgroup
of
so that
is isomorphic to
.
In the literature about sporadic groups wordings like „
is involved in
“[1] can be found with the apparent meaning of „
is a subquotient of
“.
As in the context of subgroups, in the context of subquotients the term trivial may be used for the two subquotients
and
which are present in every group
.[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]
Example
There are subquotients of groups which are neither subgroup nor quotient of it. E. g. according to 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
. It shall be proved for groups.
- Notation
- For group
, subgroup
of
and normal subgroup
of
the quotient group
is a subquotient of
, i. e.
.
- Reflexivity:
, i. e. every element is related to itself. Indeed,
is isomorphic to the subquotient
of
. - Antisymmetry: if
and
then
, i. e. no two distinct elements precede each other. Indeed, a comparison of the group orders of
and
then yields
from which
. - Transitivity: if
and
then
.
Proof of transitivity for groups
Let
be subquotient of
, furthermore
be subquotient of
and
be the canonical homomorphism. Then all vertical (
) maps
| | ![{\displaystyle G''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a37ec377e9bb29c7dd95a844c1b230fbbebea75) | ![{\displaystyle \leq }](https://wikimedia.org/api/rest_v1/media/math/render/svg/440568a09c3bfdf0e1278bfa79eb137c04e94035) | ![{\displaystyle \varphi ^{-1}(H'')}](https://wikimedia.org/api/rest_v1/media/math/render/svg/abdbf35244e79141c39480bfd0c870686ef8c841) | ![{\displaystyle \leq }](https://wikimedia.org/api/rest_v1/media/math/render/svg/440568a09c3bfdf0e1278bfa79eb137c04e94035) | ![{\displaystyle \varphi ^{-1}(H')}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05bb9679df63557dd17ce412bfe31527544c1ba8) | ![{\displaystyle \vartriangleleft }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d9d6e302fa02d3bb6a353a548cc173fd2cc3b90) | ![{\displaystyle G'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76634fad5818a777669a77cd8c86d1d816e4c402) | |
| ![{\displaystyle \varphi \!:}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dff1165f3a61965dc4d8c380a2de6dfab0e528cb) | ![{\displaystyle {\Big \downarrow }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33194d9036970c4980c32f4293bec52fa4a3edc4) | | ![{\displaystyle {\Big \downarrow }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33194d9036970c4980c32f4293bec52fa4a3edc4) | | ![{\displaystyle {\Big \downarrow }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33194d9036970c4980c32f4293bec52fa4a3edc4) | | |
| | ![{\displaystyle \{1\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5acdcac635f883f8b4f0a01aa03b16b22f23b124) | ![{\displaystyle \leq }](https://wikimedia.org/api/rest_v1/media/math/render/svg/440568a09c3bfdf0e1278bfa79eb137c04e94035) | ![{\displaystyle H''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ddcdb42f04e20193efceffe1e6bfefc2835bbf54) | ![{\displaystyle \vartriangleleft }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d9d6e302fa02d3bb6a353a548cc173fd2cc3b90) | ![{\displaystyle H'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/575e08b1574dc1c2bb8c5941a2a68d6daca7fd8e) | ![{\displaystyle \vartriangleleft }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d9d6e302fa02d3bb6a353a548cc173fd2cc3b90) | |
are surjective for the respective pairs
| ![{\displaystyle (X,Y)\;\;\;\in }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c64ca7167bae328e705cea185f12418a3f82939) | ![{\displaystyle {\Bigl \{}{\bigl (}G'',\{1\}{\bigr )}{\Bigr .}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c24865ee05517bf69f3b4bb35f8a5d3e6c45ae1b) | ![{\displaystyle ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8fa4cba3a446de313920e16251756e27312b825) | ![{\displaystyle {\bigl (}\varphi ^{-1}(H''),H''{\bigr )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b7ce691f95bbbcc54784fbe66e1e6d8bdccdc18) | ![{\displaystyle ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8fa4cba3a446de313920e16251756e27312b825) | ![{\displaystyle {\bigl (}\varphi ^{-1}(H'),H'{\bigr )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bcbb9137429b0e49759256b3fe725a7818d6971b) | ![{\displaystyle ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8fa4cba3a446de313920e16251756e27312b825) | |
The preimages
and
are both subgroups of
containing
and it is
and
because every
has a preimage
with
Moreover, the subgroup
is normal in
As a consequence, the subquotient
of
is a subquotient of
in the form
Relation to cardinal order
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
of
is either the empty set or there is an onto function
. This order relation is traditionally denoted
If additionally the axiom of choice holds, then
has a one-to-one function to
and this order relation is the usual
on corresponding cardinals.
See also
References
- ^ 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