Generalizations of Lagrange and Sylow Theorems for Groupoids
Gustav Beier, Christian Garcia, Wesley G. Lautenschlaeger, Juliana Pedrotti, Thaísa Tamusiunas
aa r X i v : . [ m a t h . R A ] J a n GENERALIZATIONS OF LAGRANGE AND SYLOWTHEOREMS FOR GROUPOIDS
GUSTAV BEIER, CHRISTIAN GARCIA, WESLEY G. LAUTENSCHLAEGER,JULIANA PEDROTTI AND THA´ISA TAMUSIUNAS
Abstract.
We show a classification method for finite groupoids anddiscuss the cardinality of cosets and its relation with the index. Weprove a generalization of the Lagrange’s Theorem and establish a Sylowtheory for groupoids.
Keywords: groupoid, Lagrange’s theorem, Sylow theorems1.
Introduction
On group theory, Lagrange’s Theorem states that, given a finite group G ,the order of any subgroup H divides the order of G . Precisely, it establishesthat the number of cosets of H in G is given by the order of G divided bythe order of H .A groupoid is usually presented as a small category whose morphisms areinvertible, which is a natural extension of the notion of group. Indeed, anygroup can be seen as a category with a unique object. An algebraic interpre-tation of groupoids appeared for the first time in [5], but a generalization ofgroup theory for the case of groupoids took a while to be studied. A Cayleytheorem for groupoids appeared in [6]. A theory for normal subgroupoidand quotient groupoid was given in [7]. Normal ordered subgroupoids andquotient ordered groupoids were studied in [1]. In [2] isomorphism theo-rems for groupoids were proved, such as results of normal and subnormalgroupoid series. In addition, in [3] the notions of center, commutator andinner isomorphism for groupoids were presented.Our main goal in this paper is to prove a Lagrange’s Theorem for groupoidsand to show some of its direct consequences in the generalization of grouptheory. We also extend the Sylow theory, which guarantees the existence ofsubgroupoids of a given order, as well as some properties about them.This work is organized as it follows. In section 2 we provide a backgroundabout groupoids and fix some notations. In section 3 we determine the orderof a finite subgroupoid in terms of its connected components and we presentthe first part of Lagrange’s Theorem. Also, we give a method to classify finitegroupoids. In section 4 we discuss about cosets and its relations with the index and we prove a generalization of the Lagrange’s Theorem for groupoids.The last section will address a generalizaton of Sylow theory.2. Preliminaries
Throughout this paper we adopt the algebraic definition of a groupoid,which appears, for instance, in [7]. This approach is completely equivalentto its categorical definition. A groupoid G is a nonempty set, equipped witha partially defined binary operation, which we will denote by concatenation,that satisfies the associative law (whenever it makes sense) and the conditionthat every element g ∈ G has an inverse g − and a right and a left identity,respectively denoted by d ( g ) and r ( g ) and named domain and range of g . Itis immediate to check that the composition gh of two elements of G existsif and only if d ( g ) = r ( h ). Also, G will denote the set of identities of G .A subgroupoid of G is a nonempty subset H , equipped with the restrictionthe of operation of G , that is a groupoid itself. We say that H is wide if H = G .Given e ∈ G , we write the set G e = { g ∈ G : r ( g ) = d ( g ) = e } . It iseasy to verify that G e is a group for all e ∈ G , called the isotropy groupassociated with e . The isotropy subgroupoid of G is defined asIso( G ) = · [ e ∈G G e . A groupoid G is said to be connected if given any e , e ∈ G there exists g ∈ G with d ( g ) = e and r ( g ) = e . In a connected groupoid, all theisotropy groups are isomorphic.Given e , e ∈ G we set G ( e , e ) := { g ∈ G : d ( g ) = e and r ( g ) = e } . It is well-known that any groupoid is a disjoint union of connectedsubgroupoids. Indeed, we define the following equivalence relation on G :for all e , e ∈ G , e ∼ e if and only if G ( e , e ) = ∅ . Every equivalence class ¯ e ∈ G / ∼ determines a connected subgroupoid G ¯ e of G , whose set of identities is ¯ e . The subgroupoid G ¯ e is called the connectedcomponent of G associated to ¯ e . It is clear that G = ˙ ∪ ¯ e ∈G / ∼ G ¯ e .Given a nonempty set X , we recall that the coarse groupoid associatedto X is the groupoid X = X × X , where ∃ ( x, y )( u, v ) if and only if x = v and in this case ( x, y )( u, v ) = ( u, y ). The identities of X are the elementsof the form ( x, x ) and the inverse element of the pair ( x, y ) is given by( x, y ) − = ( y, x ). If G is a connected groupoid, [4, Proposition 2.1] relates itclosely to its isotropy groups. Under this condition, we have G ≃ G × G e ,and this does not depend on the choice of e ∈ G . Notice that every two finitecoarse groupoids with the same amount of identities are isomorphic. Hencewe will just denote by A n the coarse groupoid with n identities. Thus, if G isa finite connected groupoid with |G | = k and e ∈ G , we have G ≃ A k × G e . ENERALIZATIONS OF LAGRANGE AND SYLOW THEOREMS 3 Classification of finite groupoids
Given a finite groupoid, the purpose of this section is to determine theorder of a subgroupoid in terms of its connected components and underwhat conditions this order divides the order of the entire groupoid. This isthe first part of Lagrange’s Theorem. Also, we want to use this descriptionto classify finite groupoids. We start by studying the case of connectedgroupoids and then move forward to the general case.Fix the notation |G| for the order of a groupoid G . From now on, G willalways denote a finite groupoid with k identities. Proposition 3.1.
Let H be a connected subgroupoid of a connected groupoid G and e ∈ H . If |H | = d , then H ≃ A d × K , where K is a subgroup of G e .Proof. It is straightforward. (cid:3)
We study now the case where H is not necessarily connected. Example 3.2.
Let H be a subgroupoid of a connected groupoid G . Supposethat H has two connected components K and L . Hence K and L are con-nected subgroupoids of G . Thus, by Proposition 3.1, if d = |K | , d = |L | , e ∈ K and f ∈ L , we obtain K ≃ A d × K e and L ≃ A d × L f . Since K e , L f are subgroups of G e ≃ G f , it follows that |K e | and |L f | divide |G e | .Hence |H| = |K| + |L| = d · |K e | + d · |L f | . With a simple exercise of computation, the next proposition extends theexample above.
Proposition 3.3.
Let H be a subgroupoid of a connected groupoid G withconnected components K , K , . . . , K m and let e i ∈ ( K i ) for all i ∈ { , , . . . , m } .Denote by k i = | ( K i ) | . Then |H| = m X i =1 k i · | ( K i ) e i | . The next result is the first part of the Lagrange’s Theorem. The completetheorem will be proved in the next section.
Theorem 3.4.
Let G be a groupoid with connected components G i , ≤ i ≤ t , e i ∈ ( G i ) and H a subgroupoid of G . The following statements hold: (i) the order of H is of the form |H| = ℓ X i =1 d i m i , where ℓ is the amount of connected components of H , P d i ≤ k and m i divides |G e i | . (ii) if G and H are connected and |H | divides |G | , then |H| divides |G| . BEIER, GARCIA, LAUTENSCHLAEGER, PEDROTTI AND TAMUSIUNAS
Proof. (i) Denote by H i = G i ∩H and K i , . . . , K n i i the connected componentsof H i . Notice that H i is a subgroupoid of G i , for 1 ≤ i ≤ t . Let l be suchthat H i = G i ∩ H 6 = ∅ , for 1 ≤ i ≤ l . Therefore, by Proposition 3.3, |H i | = n i X j =1 ( k ji ) · | ( K ji ) e ji | , where k ji = | ( K ji ) | , e ji ∈ ( K ji ) and | ( K ji ) e ji | divides |G e i | , for all 1 ≤ i ≤ l and 1 ≤ j ≤ n i . Since the connected components of G are disjoint, it followsthat |H| = l X i =1 |H i | = l X i =1 n i X j =1 ( k ji ) · (cid:12)(cid:12)(cid:12) ( K ji ) e ji (cid:12)(cid:12)(cid:12) . The statement P i,j k ji ≤ k now is obvious since k ji = (cid:12)(cid:12)(cid:12) ( K ji ) (cid:12)(cid:12)(cid:12) . In fact, ∪ i,j ( K ji ) = H ⊆ G , from where it follows that P i,j k ji = (cid:12)(cid:12)(cid:12) ∪ i,j ( K ji ) (cid:12)(cid:12)(cid:12) ≤|G | = k .(ii) Denote by d = |H | . Thus H = A d × H e , for any e ∈ H . Since |H e | divides |G e | and d divides k , we have that |H| = d · |H e | divides k · |G e | = |G| . (cid:3) The results above show that the order of a finite groupoid G is of the form |G| = n m + · · · + n ℓ m ℓ , where each n i m i is the order of a connected component of G such that n i is the amount of identities of the i -th connected component and m i is theorder of the isotropy group of some identity of the i -th connected component.From that we can obtain some interesting results. Corollary 3.5.
Every connected groupoid with order p p · · · p n is a group,where the p i ’s are distinct primes.Proof. Assume that G is not a group. Then p p · · · p n = k m , where k = 1,which is a contradiction. (cid:3) Given an order n , we have a method to find out how many groupoidsof order n there are. It is an inductive process, and we use it to classifygroupoids using the classification of finite groups and of smaller connectedgroupoids.The classification of a groupoid of order n relies on the number of itsconnected components. In number theory and combinatorics, a partition of n is a way to write it as a sum of positive integers; two sums that differ onlyin the position of their terms are considered the same partition. A summandof a partition is called a part . Using these notations, the problem can betranslated as classifying connected groupoids. ENERALIZATIONS OF LAGRANGE AND SYLOW THEOREMS 5
Fixed the order n and the number of connected components t , we needto write all the partitions t X i =1 a i = n of n , where each part is given by a i = n i m i , with n i , m i ∈ N and P n i = k .At last, we must classify each part as a connected groupoid using thedecomposition A n i × G i , where G i is a group of order m i . We will classifyall groupoids with order between 1 and 6 using this method in the nextexample. Example 3.6. If G is a groupoid of order 1, 2 or 3, then G is a group ora disjoint union of groups. As all groups of order 1, 2 or 3 are cyclic, thenumber of groupoids of order 1, 2, 3 are, respectively, 1, 2 and 3.If G is a groupoid of order 4, G is a group - Z or K , the Klein group- a disjoint union of groups or G = A . The groupoid A is the smallestgroupoid which is not a disjoint union of groups. There are 7 groupoids oforder 4.If |G| = 5, the number of partitions of 5 is 7: • For 5, G has one connected component, then G ≃ Z ; • For 4 + 1, we have the union of disjoint connected groupoids, with 3options for the connected component of order 4; • For 3 + 2, we have the disjoint union of Z and Z ; • For 3 + 1 + 1, 2 + 2 + 1, 2 + 1 + 1 + 1 and 1 + 1 + 1 + 1 + 1, we havemore disjoint unions of groups.Then, we can see that there are 8 groupoids of order 5.If G is a groupoid of order 6, G has at least one connected component andat most six connected components. If G has only one connected component, G is a group, since 6 = 2 ·
3, two distinct primes. There are two groupsof order 6: Z , the cyclic group of order 6, and D , the dihedral group ofdegree 3.If G has two connected components, we have: • For 5 + 1, we have the disjoint union of Z and the trivial group. • For 4 + 2, we have the disjoint union of Z and of a connectedgroupoid of order 4. There are three options: Z , K and A . • For 3 + 3, we have the disjoint union of Z and Z .If G has three connected components, for 4 + 1 + 1, there are three options.All the other groupoids will have connected components with order lessthan 4. For groupoids with four, five or six connected components, we haveanother 6 options, which are the disjoint union of groups. Therefore, thereare 16 non-isomorphic groupoids of order 6. BEIER, GARCIA, LAUTENSCHLAEGER, PEDROTTI AND TAMUSIUNAS Index, Cosets and Lagrange’s Theorem
In this section we will establish relations between the cardinality of thecosets of a finite groupoid and its order, and use it to generalize the La-grange’s Theorem. For that, if H is a subgroupoid of G the relation ≡ H in G is defined in [7] as x ≡ H y ⇔ ∃ yx − and yx − ∈ H . For g ∈ G we define the right coset of H in G that contains g by H g = { hg : h ∈ H and d ( h ) = r ( g ) } . We can define left cosets in a similar way. Besides that, there is a bijectionbetween the sets of left and right cosets given by H g g − H . Thereforewe can denote the amount of cosets of H in G by ( G : H ).In the case of groups, we have that ( G : H ) | H | = | G | , because every cosethas the same cardinality and every element belongs to a coset. This is notthe case of groupoids. Example 4.1.
Take the groupoid G = A × S , where S = { , σ } andthe subgroupoid H = { e , e , g , g } × S ∪ { e } of G as in the followingsimplified diagram: e e e S We can observe that the cosets g H = { g } and g H = { g , g σ, e ,g g σ } do not have the same cardinality.Consider K = { e , e , g , g } × S . Notice that g ∈ g K while K g = ∅ . If g ∈ G is such that g H = ∅ or H g = ∅ we will simply write that thecoset of g in H is empty. In fact, every element belongs to a coset if andonly if H is a wide subgroupoid of G . Lemma 4.2.
Let g ∈ G . If r ( g ) ∈ H , denote by H ∗ the connected componentof H such that r ( g ) ∈ H ∗ . Define δ = ( |H ∗ | , if r ( g ) ∈ H , if r ( g ) / ∈ H . Then |H g | = δ · |H r ( g ) | .Proof. Notice that H g = { hg : h ∈ H , d ( h ) = r ( g ) } . ENERALIZATIONS OF LAGRANGE AND SYLOW THEOREMS 7
Since the cancellation law is valid in the case of groupoids, we have that |H g | = |{ hg : h ∈ H , d ( h ) = r ( g ) }| = |{ h ∈ H : d ( h ) = r ( g ) }| = |{ h ∈ H ∗ : d ( h ) = r ( g ) }| . Now we can use a simple couting argument and conclude that |{ h ∈ H ∗ : d ( h ) = r ( g ) }| = δ · |H r ( g ) | . (cid:3) Lemma 4.3.
Let H be a subgroupoid of a connected groupoid G , H and H two connected components of H , f ∈ G \ H and G ′ = { g ∈ G : r ( g ) ∈ ( H ) and d ( g ) = f ∈ ( H ) } . Then the amount of cosets of the type H g where g ∈ G ′ is | ( H ) | ( G e : ( H ) e ) . Proof.
By Proposition 3.1 we have that
G ≃ A k ×G e and H ≃ A | ( H ) | ×H e . Thus the elements g ∈ G ′ can be seen as (( f, r ( g )) , g ′ ) ∈ A | ( H ) | × G e where g ′ ∈ G e . Hence H g ≃ ( A | ( H ) | × H e )(( f, r ( g )) , g ′ )= A | ( H ) | ( f, r ( g )) × H e g ′ = { (( a, a )( f, r ( g )) , hg ′ ) : a ∈ H and h ∈ H e } . Therefore the amount of cosets of the type H g , for fixed f ∈ ( H ) , isdetermined only by the amount of cosets of the isotropy group. Hence, thenumber of cosets is ( G e : ( H ) e ) . Since there are | ( H ) | distinct identities,the amount of cosets of the type H g where g ∈ G ′ is precisely given by | ( H ) | ( G e : ( H ) e ). (cid:3) Theorem 4.4.
Let H be a subgroupoid of a connected groupoid G with con-nected components denoted by H i and, for e i ∈ ( H i ) , the isotropy group H e i of H i for, ≤ i ≤ n . Then ( G : H ) = |H | n X i =1 ( G e : H e i ) ! . Proof.
The proof will be given by induction in the number of connectedcomponents of H and in the number of identities of H . Assume that H isconnected with only one identity. Given g ∈ G , the coset H g is not empty ifand only if there is h ∈ H such that r ( g ) = d ( h ). Hence ( G : H ) = ( G e : H e ).Now, assume that for any subgroupoid H which is connected and has l identities, the equality ( G : H ) = l ( G e : H e )holds. BEIER, GARCIA, LAUTENSCHLAEGER, PEDROTTI AND TAMUSIUNAS
For the induction step, suppose that H is connected and has l +1 identitiesdenoted by { e , . . . , e l +1 } . Consider the subgroupoid H ′ = { g ∈ H : d ( g ) = e i and r ( g ) = e j , for 1 ≤ i, j ≤ l } . By the induction assumption we havethat ( G : H ′ ) = l ( G e : H e ).Notice that if g ∈ G is such that d ( g ) = e i and r ( g ) = e j , for 1 ≤ i, j ≤ l +1,then the coset of g in H is empty. If d ( g ) = e l +1 and r ( g ) = e j , for 1 ≤ j ≤ l or d ( g ) = e i , for 1 ≤ i ≤ l , and r ( g ) = e l +1 , the coset of g in H wasalready counted by ( G : H ′ ). Therefore, it only remains to us to observewhat happens to the elements g ∈ G such that d ( g ) = r ( g ) = e l +1 , thatgenerate ( G e : H e ) additional cosets. Hence( G : H ) = ( G : H ′ ) + ( G e : H e ) = l ( G e : H e ) + ( G e : H e ) = ( l + 1)( G e : H e ) , which concludes the induction in the number of identities.Assume now that for any subgroupoid H which has m connected compo-nents and l identities, the equality( G : H ) = |H | m X i =1 ( G e : H e i ) ! holds. Suppose that H has m + 1 connected components and l identities.By induction assumption, the subgroupoid H ′ = ∪ mi =1 H i is such that( G : H ′ ) = |H ′ | m X i =1 ( G e : H e i ) ! which is the number of cosets of elements g ∈ G such that d ( g ) and r ( g ) arein H ′ .Notice that H m +1 is a connected component, so the first part of the proofsays that the cosets of g ∈ G such that d ( g ) ∈ H m +1 and r ( g ) ∈ H m +1 arecounted as | ( H m +1 ) | ( G e : H e m +1 ) . If g ∈ G is such that d ( g ) ∈ H m +1 and r ( g ) ∈ H i , for some 1 ≤ i ≤ m ,the number of cosets of the type H g is | ( H m +1 ) | ( G e : H e i ) | by Lemma 4.3.Since there are m connected components we have that the number of cosetsgenerated by { g ∈ G : d ( g ) ∈ H m +1 and r ( g ) ∈ H i } is | ( H m +1 ) | m X i =1 ( G e : H e i ) ! . If g ∈ G is such that d ( g ) ∈ H i , for some 1 ≤ i ≤ m and r ( g ) ∈ H m +1 , thenumber of cosets of the type H g is ( G e : H e m +1 ) by Lemma 4.3. Since thereare m connected components we have that the number of cosets generated ENERALIZATIONS OF LAGRANGE AND SYLOW THEOREMS 9 by { g ∈ G : d ( g ) ∈ H i and r ( g ) ∈ H m +1 } is |H ′ | ( G e : H e m +1 ). Hence( G : H ) = |H ′ | m X i =1 ( G e : H e i ) ! + | ( H m +1 ) | m X i =1 ( G e : H e i ) ! + |H ′ | ( G e : H e m +1 ) + | ( H m +1 ) | ( G e : H e m +1 )= |H | m X i =1 ( G e : H e i ) ! + |H | ( G e : H e m +1 )= |H | m +1 X i =1 ( G e : H e i ) ! . (cid:3) Theorem 4.5.
Let G j be the connect components of G and G e j the isotropygroup of G j for ≤ j ≤ t , and let H be a subgroupoid of G with connectedcomponents denoted by K i and, for e i ∈ ( K i ) , K e i the isotropy group of K i ,for ≤ i ≤ n and H j = G j ∩ H . Then ( G : H ) = t X j =1 | ( H j ) | n X i =1 ( G e i : K e i ) ! . Proof.
Just apply the Theorem 4.4 in each connected component of G . (cid:3) Theorem 4.6.
Let G be a connected groupoid and H a wide subgroupoidwith connected components H i and, for e i ∈ ( H i ) , H e i the isotropy groupof H i for ≤ i ≤ n . Then |G| = |G | n X i =1 ( G e i : H e i ) | ( H i ) ||H e i | ! . Proof.
We have that |G| = |G | |G e | = |G | n X i =1 | ( H i ) | ! |G e | = |G | n X i =1 | ( H i ) ||G e | ! = |G | n X i =1 | ( H i ) ||G e i | ! = |G | n X i =1 | ( H i ) | ( G e : H e i ) |H e i | ! , which concludes the proof. (cid:3) Notice that if H is a subgroup of a finite group G , Theorem 4.6 preciselystates that | G | = ( G : H ) | H | . The condition of G being a connected groupoidcan be disregarded, as we can see below.The next result is the second part of the Lagrange’s Theorem. Theorem 4.7.
Let G j be the connected components of G and G e j the isotropygroup of G j for ≤ j ≤ t , let H be a wide subgroupoid of G with connectedcomponents K i and, for e i ∈ ( K i ) , let K e i be the isotropy group of K i , for ≤ i ≤ n and H j = G j ∩ H . Then |G| = t X j =1 | ( G j ) | n X i =1 ( G e j : K e i ) | ( H j ) ||K e i | ! . Proof.
Just apply the Theorem 4.6 in each connected component of G . (cid:3) Using Theorem 3.4 and Theorem 4.7, we can finally state a generalizationof the Lagrange’s Theorem, without concern about the connectedness of thegroupoid.
Theorem 4.8 (Lagrange’s Theorem for Groupoids) . Let G be a groupoidwith connected components G j , ≤ j ≤ t , e j ∈ ( G j ) and H a subgroupoidof G . The following statements hold: (i) the order of H is of the form |H| = ℓ X j =1 d j m j , where ℓ is the amount of connected components of H , P d j ≤ k and m j divides |G e j | . (ii) if G and H are connected and |H | divides |G | , then |H| divides |G| . (iii) if G e j is the isotropy group of G j , H is wide with connected compo-nents K i and, for e i ∈ ( K i ) , K e i is the isotropy group of K i , for ≤ i ≤ n and H j = G j ∩ H , then |G| = t X j =1 | ( G j ) | n X i =1 ( G e j : K e i ) | ( H j ) ||K e i | ! . Corollary 4.9.
Let G be a connected groupoid and H be a wide subgroupoidwith connected components H i and isotropy groups H e i , e i ∈ ( H i ) , suchthat | ( H i ) | and |H e i | are the same for all ≤ i ≤ n . Then |G| = ( G : H ) | ( H i ) ||H e i | . ENERALIZATIONS OF LAGRANGE AND SYLOW THEOREMS 11
Proof.
Using the Theorems 4.4 and 4.6, we obtain |G| = |G | n X i =1 ( G e i : H e i ) | ( H i ) ||H e i | ! = |G | n X i =1 ( G e i : H e i ) ! | ( H i ) ||H e i | = ( G : H ) | ( H i ) ||H e i | . (cid:3) Sylow Theorems
The purpose of this section is to answer the following question: given aconnected groupoid G with |G | = k and d , d , . . . , d ℓ , m , m , . . . , m ℓ ∈ N with P i d i ≤ k and m i a divisor of |G e | for all i , when is there a subgroupoid H of G with order d m + d m + · · · + d ℓ m ℓ ?During this section we will consider G a connected groupoid. The discon-nected case can be obtained by applying the results of this section to eachconnected component of the groupoid. Our first result is a consequence ofthe First Sylow Theorem for groups. Lemma 5.1.
Consider e ∈ G and p a prime such that |G e | = p m b , with gcd( p, b ) = 1 . Then for all ≤ n ≤ m there is a wide connected subgroupoid H of G such that |H| = k p n . In particular, |H| divides |G| .On the other hand, if ≤ d ≤ k , there is a connected subgroupoid H of G such that |H | = d and |H| = d p n , for all ≤ n ≤ m .Proof. Since G is connected, G ≃ A k × G e . We know that G e is a group, sowe can use the First Sylow Theorem for groups to obtain a subgroup K of G e with order p n for all 0 ≤ n ≤ m .Consider H = A k × K . It is evident that H is a wide connected sub-groupoid of G . Therefore |H| divides |G| by the Theorem 4.8. Besides that, H e is precisely K , since K ⊆ G e . Thus |H e | = | K | = p n .For the second statement, consider a connected subgroupoid G ′ = A d × G e of G and use the argument above to obtain a wide connected subgroupoid H of G ′ . The result is now direct. (cid:3) Once d and p above are fixed, inspired by the group notation, we will calla subgroupoid H obtained by ( d, p )-subgroupoid of G . When |H e | = p m ,we denote by ( d, p )-Sylow subgroupoid. If d = k , we will write only p -subgroupoid or p -Sylow subgroupoid.The Second Sylow Theorem for groups tells us how the p -Sylow subgroupsare related. We will now generalize this result for the case of connectedgroupoids. Before that, we will recall the definition of normal subgroupoidand define characteristic subgroupoid. Definition 5.2.
Let G be a (not necessarily finite or connected) groupoidand H a wide subgroupoid of G . (i) We say that H is normal and denote H ⊳ G if g − H g ⊆ H for all g ∈ G .(ii) Define A ( G ) = { f : G e → G e ′ : e, e ′ ∈ G and f is an isomorphism } . We say that H is characteristic and denote H ◭ G if H is invariantunder all elements of A ( G ). That is, if f ( H ∩ dom( f )) = H ∩
Im( f ).It is clear that every characteristic subgroupoid is normal. In fact, a sub-groupoid is normal if and only if it is invariant under the inner isomorphismsof G , that together are a subset of A ( G ) [3, Proposition 5.2]. Notice that A ( G ) = Aut( G ) in general. Example 5.3.
It is evident that G ◭ G and G ◭ G . Defining Z ( G ) = { g ∈ Iso( G ) : gh = hg for all h ∈ G d ( g ) } , the center of G [3, Definition 4.1], we have that Z ( G ) ◭ G .In fact, take f ∈ A ( G ). Consider f : G e → G e ′ . Then Z ( G ) ∩ G e = Z ( G e ).Hence, f ( Z ( G ) ∩ G e ) = f ( Z ( G e )). We will show that f ( Z ( G e )) = Z ( G e ′ ) = Z ( G ) ∩ G e ′ . Let g ′ ∈ f ( Z ( G e )). Thus there is g ∈ Z ( G e ) such that g ′ = f ( g ) and r ( g ′ ) = e ′ = d ( g ′ ). Let h ′ ∈ G e ′ . Since f is an isomorphism, there is h ∈ G e such that f ( h ′ ) = h . Hence g ′ h ′ = f ( g ) f ( h ) = f ( gh ) = f ( hg ) = f ( h ) f ( g ) = h ′ g ′ , because g ∈ Z ( G e ). Thus g ′ ∈ G e ′ as we wanted. The other inclusion is givensimilarly since f is an isomorphism.A classic result in group theory is that the relation ⊳ is not transitive,but can be repaired with the relation ◭ . This result is also true in groupoidtheory. Proposition 5.4.
Let G be a groupoid and let H , K be subgroupoids of G such that K ◭ H ⊳ G . Then K ⊳ G .Proof. Consider I g : G d ( g ) → G r ( g ) the partial inner isomorphism of G givenby I g ( x ) = gxg − . Since H is normal, we have that I g ( H d ( g ) ) = I g ( H ∩ G d ( g ) ) = H ∩ G r ( g ) = H r ( g ) . Hence I g | H : H d ( g ) → H r ( g ) is an element of A ( G ). Since K is characteris-tic, I g ( K ∩ G d ( g ) ) = I g ( K ∩ H d ( g ) )= I g | H ( K ∩ H d ( g ) )= K ∩ H r ( g ) = K ∩ G r ( g ) . This shows us that K is invariant under every partial inner isomorphismof G , that is, K is normal. (cid:3) ENERALIZATIONS OF LAGRANGE AND SYLOW THEOREMS 13
Definition 5.5.
Denote by W C ( G ) the set of all wide connected subgroupoidsof G and let H ∈ W C ( G ). Define I g : W C ( G ) → W C ( G ) H ≃ A k × H d ( g )
7→ A k × g H d ( g ) g − . We will call the wide connected subgroupoid I g ( H ) by isotropic conjugate of H . Lemma 5.6.
Let p be a prime, ≤ d ≤ k and n d,p the number of ( d, p ) -Sylow subgroupoids of G . (i) All ( d, p ) -Sylow subgroupoids of G are isotropic conjugates. In par-ticular, a p -Sylow subgroupoid H is such that H ⊳ G if and only if n k,p = 1 . In this case, H ◭ G . (ii) If P is a ( d, p ) -subgroupoid of G , there is a ( d, p ) -Sylow subgroupoid S of G such that P ⊆ S . (iii) If S is a p -Sylow subgroupoid, we have that n k,p = ( G e : N G e ( S e )) ,for any e ∈ G .Proof. (i): The statement is equivalent to show that given H , K ( d, p )-Sylowsubgroupoids of G , e ∈ H and f ∈ K , we have that there is g ∈ G suchthat g H e g − = K f .Since G is connected, there is x ∈ G with r ( x ) = f , d ( x ) = e . We alreadyknow that G e ≃ G f via I x . Thus I x ( H e ) = x H e x − is a p -Sylow subgroup of G f . Since all p -Sylow subgroups of a group are conjugates, there is y ∈ G f such that yx H e x − y − = K f . Therefore g = yx is the element that wewanted.On the other hand, let H be a ( d, p )-Sylow subgroupoid. It is obvious that I g ( H e ) = g − H e g is a p -Sylow subgroup of G f . Therefore we can construct asuitable ( d, p )-Sylow subgroupoid K = A d × g − H e g such that K f = g − H e g .Now, notice that H ⊳ G if and only if g − H g ⊆ H , for all g ∈ G . We havethat g − H g = g − H r ( g ) g is precisely an isotropy group for some p -Sylowsubgroupoid of G . Since H is unique, we have that g − H r ( g ) g = H d ( g ) ⊆ H proving that H is normal. For the converse, notice that if H is normalthen g − H r ( g ) g = H d ( g ) for all g ∈ G . But this implies that every p -Sylowsubgroupoid of G has the exact same elements as H . Thus, H must beunique.In fact, we have proved even more: a p -Sylow subgroupoid H is normal in G if and only if every p -Sylow subgrpup H e is normal in G e for all e ∈ G if andonly if every p -Sylow subgroup H e is characteristic in G e for all e ∈ G . Thelast equivalence follows from the Second Sylow Theorem for groups. Sincethe definition of characteristic subgroupoid depends only on the isotropysubgroups, it follows that H ◭ G .(ii): If P is a ( d, p )-subgroupoid of G , then P e is a p -subgroup of G e , forall e ∈ P . Hence there is S e ⊆ G e p -Sylow subgroup for all e ∈ P . Nowtake S = A d × S e . (iii): By the Second Sylow Theorem for groups we have that G e has( G e : N G e ( S e )) p -Sylow subgroups. The result follows directly from (i). (cid:3) Example 5.7.
Let G = A × D , where D = { , ρ, ρ , τ , τ , τ } is thedihedral group of degree 3. We know that |G| = 3 · h τ i i , 1 ≤ i ≤ , and only one 3-Sylow subgroupgiven by h ρ i . Since |G | = 3, we have the options d = 1, d = 2 or d = 3.For d = 1, take A × H ≃ H , where H is a Sylow subgroup of D . Wehave that n , = 3 · n , = 3 ·
3, given the choice of the identity e i ∈ G and of the Sylow subgroup of D . The (1 , G are { e } × h ρ i , { e } × h ρ i and { e } × h ρ i . For d = 2, taking A × H we obtain n , = 3 · n , = 3 ·
3, sincewe have three distinct but isomorphic coarse groupoids A in A . Denotingby A ij the coarse subgroupoid of A that contains e i , e j , it follows that thethree (2 , G are isotropic conjugates of the form A ij × h ρ i , and have order 12. The nine (2 , A ij ×h τ k i have order 8. Observe that in this case the orders of the subgroupoids donot divide the order of G .Taking d = 3, we have the wide Sylow subgroupoids of G . Thus, n , = 1and n , = 3. Now observe that the (3 , A × h ρ i is unique and therefore normal. On the other hand, the (2 , A × h τ i i are not normal since they are isotropic conjugated.The next lemma will give us more information about the number n d,p . Lemma 5.8.
Let p be a prime and let e ∈ G be such that |G e | = p m b with gcd( p, b ) = 1 . Then n d,p = N · (cid:18) kd (cid:19) , where N is such that ( N divides b,N ≡ p. Proof.
Consider N = n p the number of p -Sylow subgroups of G e . By theThird Sylow Theorem for groups, N | b and N ≡ p .Notice that the Sylow subgroupoids of G are of the form A d × H where H is a p -Sylow subgroup of G e and A d ⊂ A k . By a counting exercise we cansee that there are (cid:0) kd (cid:1) isomorphic copies of A d . (cid:3) We can now state the Sylow theorems for connected groupoids.
Theorem 5.9 (First Sylow Theorem) . Let p , p , . . . , p ℓ be primes such that |G e | = p m i i b i , with gcd( p i , b i ) = 1 , for all ≤ i ≤ ℓ . Then for all ≤ n i ≤ m i ENERALIZATIONS OF LAGRANGE AND SYLOW THEOREMS 15 and ≤ d , d , . . . , d ℓ ≤ k with d = P i d i ≤ k there is a subgroupoid H of G such that |H | = d and |H| = ℓ X i =1 d i p n i i . Proof.
Just take H i as the connected ( d i , p i )-subgroupoid of G as in Lemma5.1 for all 1 ≤ i ≤ ℓ . Defining H = · S i H i we obtain the result. (cid:3) From now on, fix D = ( d , . . . , d ℓ ) , P = ( p , . . . , p ℓ ) ∈ N ℓ where P d i ≤ k and each p i is a prime as in the previous lemma. The subgroupoid abovewill be denoted by ( D, P )-subgroupoid of G . If n i = m i for all 1 ≤ i ≤ ℓ , wewill write ( D, P )-Sylow subgroupoid.
Definition 5.10.
Let H be a wide ( D, P )-Sylow subgroupoid of G , thatis, k = P i d i . Consider H i the ( d i , p i )-Sylow subgroupoid of G that is aconnected component of H . A connected components permutation of H isa subgroupoid K of G such that K ≃ H and, denoting by K i the connectedcomponent isomorphic to H i for all 1 ≤ i ≤ ℓ , there is at least one i suchthat ( K i ) = ( H i ) . Example 5.11.
As in Example 5.7, let G = A × D . Consider the disjointunion of the subgroupoids H = { e } × h ρ i and K = { e , e , g , g } × h τ i . e e e h ρ i h τ i A connected components permutation of
H∪K is the disjoint union H ′ ∪K ′ ,where H ′ = { e } × h ρ i ≃ H and K ′ = { e , e , g , g } × h τ i ≃ K . e e e h τ i h ρ i Let K ′′ = { e , e , g , g } × h τ i ≃ K ≃ K ′ . We will not consider thesubgroupoid H ′ ∪ K ′′ e e e h τ i h ρ i as a connected components permutation of H∪K , even that
H∪K ≃ H ′ ∪K ′′ ,because the isotropy group is not the same.The following lemma is a counting exercise. Lemma 5.12.
Given H a wide ( D, P ) -Sylow subgroupoid of G , there are (cid:18) kD (cid:19) := k ! d ! d ! · · · d ℓ ! connected components permutations of H . With this notation we can state the Second Sylow Theorem.
Theorem 5.13 (Second Sylow Theorem) . Let p , . . . , p ℓ be any primes, ≤ d , . . . , d ℓ ≤ k and n D,P be the number of ( D, P ) -Sylow subgroupoids of G . (i) Every two ( D, P ) -Sylow subgroupoids of G are isotropic conjugatesbesides connected components permutations. In particular, a ( D, P ) -Sylow subgroupoid H of G is normal if and only if n D,P = (cid:0) kD (cid:1) . Inthis case H and all of its connected components permutations arecharacteristic. (ii) If P is a ( D, P ) -subgroupoid of G , there is a ( D, P ) -Sylow subgroupoidof G such that P ⊆ S . (iii) If S is a ( D, P ) -Sylow subgroupoid, we have that n D,P = (cid:18) kD (cid:19) Y i ( G e i : N G ei ( S e i )) , where the S e i are the isotropy groups of each connected component S i of S for some e i ∈ ( S i ) .Proof. (i): Let H , . . . , H ℓ be the connected components of H . Define, forall 1 ≤ i ≤ ℓ , G i = A d i × G e , where e ∈ ( H i ) and A d i ⊆ A k is the coarsegroupoid whose identities are exactly the same as those of H i . We have that H i is a connected p i -Sylow subgroupoid of G i . Hence we can use Lemma5.6 to obtain that every other connected p i -Sylow subgroupoid of G i is anisotropic conjugate of H i . Thefore all ( D, P )-Sylow subgroupoids of G areisotropic conjugates besides connected components permutations.Notice that when H is normal, each H e is normal in G e . But that isthe same as saying that, once a connected components permutation is fixed,there is an unique ( D, P )-Sylow subgroupoid of G . Hence, if n D,P is exactlythe number of connected components permutations of H , every connected ENERALIZATIONS OF LAGRANGE AND SYLOW THEOREMS 17 components permutations of H is normal in G . The statement about char-acteristic subgroupoids follows directly.(ii) is evident. We will now prove (iii). Notice that n D,P = (cid:18) kD (cid:19) ℓ Y i =1 n d i ,p i , where n d i ,p i = ( G e i : N G ei ( S e i )) is the number of connected p i -Sylow sub-groupoids of G i . (cid:3) Finally, the Third Sylow Theorem gives us a system of congruences in-volving n D,P . Theorem 5.14 (Third Sylow Theorem) . Let p , . . . , p ℓ be primes and let e ∈ G be such that |G e | = p m i i b i with gcd( p i , b i ) = 1 . Then n D,P = ℓ Y i =1 N i ! · (cid:18) kD (cid:19) , where each N i is such that ( N i divides b i ,N i ≡ p i . Proof.
By the Second Sylow Theorem (iii) it follows that n D,P = (cid:18) kD (cid:19) ℓ Y i =1 n d i ,p i , where n d i ,p i is the number of connected p i -Sylow subgroupoids of G i . ByLemma 5.8, n d i ,p i = N i · (cid:18) d i d i (cid:19) = N i , where ( N i divides b i ,N i ≡ p i . (cid:3) We will conclude this work by applying the results above in the groupoid G = A × G e where |G e | = 105. Example 5.15.
Consider k = 7, D = (1 , ,
3) and P = (3 , , G is aconnected groupoid with G ≃ A × G e where |G e | = 3 · · n D,P . We already know that n D,P = (cid:18) , , (cid:19) Y i =1 N i , where N divides 35, N divides 21 and N divides 15. So N ∈ { , , , } ,N ∈ { , , , } ,N ∈ { , , , } . Hence n D,P = N · N · N · , where N = 1 or N = 7, N = 1 or N = 21 and N = 1 or N = 15,because the congruences on Theorem 5.14 hold. For more information, wewould need to know more about the group G e . For example, there are twonon-isomorphic groups of order 105.If G e = Z , we have that N = N = N = 1, since Z is abelian andall of its subgroups are normal. In this case, G is abelian in the sense of [2].So n D,P = 140 and all the 140 (
D, P )-Sylow subgroupoids of G are normalin G .Denoting by H p a p -subgroup of G e , for p ∈ { , , } , we have that n = 21and n = 15 cannot happen at the same time. In fact, we would have morethan 105 elements. So we have that n = 1 or n = 1. In either case, H = H H is normal since its index is 3. Besides that, | H | = 5 × H ≃ Z .Define a representation of some H ≃ Z ≃ h h : h = e i on H by ϕ : H × Aut( H ) → Aut( H )( h, θ ) h − θh. This will guarantee us that the semidirect product G = H ⋊ ϕ H ≃ Z ⋊ Z is well-defined and | G | = 105, so G e = G .Actually, we can prove that n = n = 1, so N = N = 1, and that n = 7, so N = 7. Then, we would have n D,P = 7 ·
140 = 980 and thesesubgroupoids would not be normal in G . References [1] N. Alyamani; N. D. Gilbert; E. C. Miller,
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ENERALIZATIONS OF LAGRANGE AND SYLOW THEOREMS 19
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