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Subquotient
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<h2 id="example">Example</h2> <p>There are subquotients of groups which are neither subgroup nor quotient of it. E. g. according to article <a href="/facts/Sporadic_group/hjNHTKVv">Sporadic group</a>, <i>Fi</i>22 has a double cover which is a subgroup of <i>Fi</i>23, so it is a subquotient of <i>Fi</i>23 without being a subgroup or quotient of it. </p> <h2 id="order-relation">Order relation</h2> <p>The relation <i>subquotient of</i> is an <a href="/facts/Partially_ordered_set/qb4a3FAX">order relation</a> – which shall be denoted by ⪯ {\displaystyle \preceq } . It shall be proved for groups. </p> Notation For <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> G {\displaystyle G} , <a href="/facts/Subgroup/r91mBgpa">subgroup</a> G ′ {\displaystyle G'} of G {\displaystyle G} ( ⇔: G ′ ≤ G ) {\displaystyle (\Leftrightarrow :G'\leq G)} and <a href="/facts/Normal_subgroup/aRDaGwLm">normal subgroup</a> G ″ {\displaystyle G''} of G ′ {\displaystyle G'} ( ⇔: G ″ ⊲ G ′ ) {\displaystyle (\Leftrightarrow :G''\vartriangleleft G')} the <a href="/facts/Quotient_group/HlwpHtjb">quotient group</a> H := G ′ / G ″ {\displaystyle H:=G'/G''} is a subquotient of G {\displaystyle G} , i. e. H ⪯ G {\displaystyle H\preceq G} . <ol><li><a href="/facts/Reflexive_relation/NzTytEg9">Reflexivity</a>: 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} .</li> <li><a href="/facts/Antisymmetric_relation/YhuqG4z8">Antisymmetry</a>: 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} .</li> <li><a href="/facts/Transitive_relation/9r90y50r">Transitivity</a>: 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} .</li></ol> <h3>Proof of transitivity for groups</h3> <p>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 <a href="/facts/Canonical_homomorphism/vBztnQUK">canonical homomorphism</a>. Then all vertical ( ↓ {\displaystyle \downarrow } ) maps φ : X → Y , x ↦ x G ″ {\displaystyle \varphi \colon X\to Y,\;x\mapsto x\,G''} </p> <table><tbody><tr><td> </td><td></td><td> G ″ {\displaystyle G''} </td><td> ≤ {\displaystyle \leq } </td><td> φ − 1 ( H ″ ) {\displaystyle \varphi ^{-1}(H'')} </td><td> ≤ {\displaystyle \leq } </td><td> φ − 1 ( H ′ ) {\displaystyle \varphi ^{-1}(H')} </td><td> ⊲ {\displaystyle \vartriangleleft } </td><td> G ′ {\displaystyle G'} </td><td></td></tr><tr><td></td><td> φ : {\displaystyle \varphi \!:} </td><td> ↓ {\displaystyle {\Big \downarrow }} </td><td></td><td> ↓ {\displaystyle {\Big \downarrow }} </td><td></td><td> ↓ {\displaystyle {\Big \downarrow }} </td><td></td><td> ↓ {\displaystyle {\Big \downarrow }} </td></tr><tr><td></td><td></td><td> { 1 } {\displaystyle \{1\}} </td><td> ≤ {\displaystyle \leq } </td><td> H ″ {\displaystyle H''} </td><td> ⊲ {\displaystyle \vartriangleleft } </td><td> H ′ {\displaystyle H'} </td><td> ⊲ {\displaystyle \vartriangleleft } </td><td> H {\displaystyle H} </td></tr></tbody></table> <p>are <a href="/facts/Surjective_function/KezhLpCb">surjective</a> for the respective pairs </p> <table><tbody><tr><td></td><td> ( X , Y ) ∈ {\displaystyle (X,Y)\;\;\;\in } </td><td> { ( G ″ , { 1 } ) {\displaystyle {\Bigl \{}{\bigl (}G'',\{1\}{\bigr )}{\Bigr .}} </td><td> , {\displaystyle ,} </td><td> ( φ − 1 ( H ″ ) , H ″ ) {\displaystyle {\bigl (}\varphi ^{-1}(H''),H''{\bigr )}} </td><td> , {\displaystyle ,} </td><td> ( φ − 1 ( H ′ ) , H ′ ) {\displaystyle {\bigl (}\varphi ^{-1}(H'),H'{\bigr )}} </td><td> , {\displaystyle ,} </td><td> ( G ′ , H ) } . {\displaystyle {\Bigl .}{\bigl (}G',H{\bigr )}{\Bigr \}}.} </td></tr></tbody></table> <p>The <a href="/facts/Preimage/KANefMAl">preimages</a> 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).} </p><p>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).} </p> <h3>Relation to cardinal order</h3> <p>In <a href="/facts/Constructive_set_theory/myLo3Mjt">constructive set theory</a>, where the <a href="/facts/Law_of_excluded_middle/tDhbugCN">law of excluded middle</a> does not necessarily hold, one can consider the relation <i>subquotient of</i> as replacing the usual <a href="/facts/Order_relation/gn6fM3oJ">order relation</a>(s) on <a href="/facts/Cardinal_number/ccoKwU4R">cardinals</a>. When one has the law of the excluded middle, then a subquotient Y {\displaystyle Y} of X {\displaystyle X} is either the <a href="/facts/Empty_set/ffEk3eA6">empty set</a> or there is an onto function X → Y {\displaystyle X\to Y} . This order relation is traditionally denoted ≤ ∗ . {\displaystyle \leq ^{\ast }.} If additionally the <a href="/facts/Axiom_of_choice/1Ura2HTh">axiom of choice</a> 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. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Homological_algebra/Jfm8w52l">Homological algebra</a></li> <li><a href="/facts/Subcountable/RVcgMVjy">Subcountable</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>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 <a href="/wiki/Robert_Griess" target="_blank">/wiki/Robert_Griess</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>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 <a href="978-0-8218-0560-2" target="_blank">978-0-8218-0560-2</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> </ol>