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 .
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
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
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![G'](https://wikimedia.org/api/rest_v1/media/math/render/svg/76634fad5818a777669a77cd8c86d1d816e4c402)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![G''](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a37ec377e9bb29c7dd95a844c1b230fbbebea75)
![G'](https://wikimedia.org/api/rest_v1/media/math/render/svg/76634fad5818a777669a77cd8c86d1d816e4c402)
![G \ geq G '\ vartriangleright G' ',](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3e500d36a4bbc755dc326c20ba40ec7c6f74fbf)
then the factor group (quotient group) is called a subquotient of .
![H: = G '/ G' '](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5cd0bfc3b39304f014109cd4fa636259267b016)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
In the literature on sporadic groups there are formulations such as
-
involved
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is involved in
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 .
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
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
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
- 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' ']}](https://wikimedia.org/api/rest_v1/media/math/render/svg/754e5b4e41f347dc4fc819e1d2c1a9425990b773)
![{\ displaystyle | G | = [G: 1] = [G: G '] \ cdot [G': G ''] \ cdot [G '': 1] = [G: G '] \ cdot | H | \ cdot | G '' |,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/302611e6717048e84db8c6419b7f15dcf7cdab8c)
and is in particular a divisor of as well![| H |](https://wikimedia.org/api/rest_v1/media/math/render/svg/31c3c97e64ad558dcc09e9ff232ee0f4d447bac1)
![| G |](https://wikimedia.org/api/rest_v1/media/math/render/svg/8258bc41edeb87bfbc8cba8367f29838c0eddc1c)
Partial order
For finite objects the relation "is subquotient of" is an order relation, namely a partial order .
is subquotient of .
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
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
![H: = G '/ G' '](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5cd0bfc3b39304f014109cd4fa636259267b016)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![\ phi \ colon G '\ to H](https://wikimedia.org/api/rest_v1/media/math/render/svg/13eb4573fb563fba764bcd78334cc58d58b0e712)
![H \ geq H '\ vartriangleright H' '](https://wikimedia.org/api/rest_v1/media/math/render/svg/9226735a3bdb7f719d3fc29f813491d391d163cf)
![H '/ H' '](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e133f56a94114a159d33fc982427bbf9512830c)
![H](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b)
![{\ 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57572036258f7b75e65dc9c3af14d809df35ef13)
with each vertical ( ) figure with matching surjective for the couple .
![\ downarrow](https://wikimedia.org/api/rest_v1/media/math/render/svg/4618f22b0f780805eb94bb407578d9bc9487947a)
![{\ displaystyle \ phi \ !: X \ to Y, g \ mapsto g \, G ''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a95e8a7a982c2907775b5312ed683d6084b38696)
![{\ displaystyle (X, Y) = {\ bigl (} G ', H {\ bigr)}, {\ bigl (} \ phi ^ {- 1} (H'), H '{\ bigr)}, { \ bigl (} \ phi ^ {- 1} (H ''), H '' {\ bigr)}, {\ bigl (} G '', \ {1 \} {\ bigr)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03b64c736346c63a2c7658908989e8e04b9571ef)
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 .
![\ phi ^ {- 1} (H ')](https://wikimedia.org/api/rest_v1/media/math/render/svg/77a1702378e899df59e38ce85ee0721786254bda)
![\ phi ^ {- 1} (H '')](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e7f603a62b4bc7763ad7bb24455557a3548e7a1)
![G'](https://wikimedia.org/api/rest_v1/media/math/render/svg/76634fad5818a777669a77cd8c86d1d816e4c402)
![G''](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a37ec377e9bb29c7dd95a844c1b230fbbebea75)
![{\ displaystyle \ phi (\ phi ^ {- 1} (H ')) = H'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2d1b0cbc6f7077767dbca94c6fb4208ffa3052f)
![{\ displaystyle \ phi (\ phi ^ {- 1} (H '')) = H ''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b301cbed9faff0499472cd862731c5711461cdb2)
![h \ in H](https://wikimedia.org/api/rest_v1/media/math/render/svg/675a79e26028d91d97f4e2ce279c314b0f194c1e)
![G'](https://wikimedia.org/api/rest_v1/media/math/render/svg/76634fad5818a777669a77cd8c86d1d816e4c402)
![\ phi ^ {- 1} (H '')](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e7f603a62b4bc7763ad7bb24455557a3548e7a1)
![\ phi ^ {- 1} (H ')](https://wikimedia.org/api/rest_v1/media/math/render/svg/77a1702378e899df59e38ce85ee0721786254bda)
![H '/ H' '](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e133f56a94114a159d33fc982427bbf9512830c)
![H](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b)
![{\ displaystyle H '/ H' '\ cong \ phi ^ {- 1} (H') / \ phi ^ {- 1} (H '')}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c42627b2a931370232ff7a7efa7974a0c730c64)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
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.
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![H](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b)
![{\ displaystyle | H | \ leq | G | \ leq | H |}](https://wikimedia.org/api/rest_v1/media/math/render/svg/391d9e310ba729b05b2c60c9d96beff193787423)
![{\ displaystyle | H | = | G |}](https://wikimedia.org/api/rest_v1/media/math/render/svg/791736612f9355979daecc462466d51892d0cc78)
![{\ displaystyle [G: G '] = 1, [G' ': 1] = 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/959f79b0c0a75930e7582919f25621b565cc9b40)
![{\ displaystyle [H: H '] = 1, [H' ': 1] = 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a913f72f99c069735c6b39738f3ab1913fefe73d)
![{\ displaystyle G \ cong H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f74c426546451af44e2fbc290e06cc084929e6b3)
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:
![\ preceq](https://wikimedia.org/api/rest_v1/media/math/render/svg/63dfe475e1377b3b4e936a3aa8fb1d7177dcdbc3)
![\ prec](https://wikimedia.org/api/rest_v1/media/math/render/svg/59707ac9078525b52a9e21a1baf9ab787af7a9aa)
- If there is one with such a way that
![{\ displaystyle I \ prec G,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4896d24cd85a5a6e2a5473ba0a758d06e0ef02f5)
![{\ displaystyle H \ prec G}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8addd45a994e0f8338ff21ff1fd8df114aa2cc19)
![{\ displaystyle I \ preceq H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1bba324bc4871f7dcff495ab74b5daa5aa131d)
Such is called a maximum real subquotient of . The term is needed , for example, when arranging the sporadic groups in the Hasse diagram .
![H](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
Individual evidence
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↑ 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.
@1@ 2Template: Webachiv / IABot / www.mathematik.uni-mainz.de
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^ Robert Griess: The Friendly Giant . In: Inventiones Mathematicae . tape 69 , 1982, pp. 91 , doi : 10.1007 / BF01389186 ( online at digizeitschriften.de ).
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↑ Noether's Isomorphy Theorems