A pastiche on embeddings into simple groups (following P. E. Schupp)
aa r X i v : . [ m a t h . G R ] F e b A PASTICHE ON EMBEDDINGS INTO SIMPLE GROUPS(FOLLOWING P. E. SCHUPP)
ZORAN ˇSUNI´C
Abstract.
Let λ be an infinite cardinal number and let C = { H i | i ∈ I } bea family of nontrivial groups. Assume that 2 ≤ | I | ≤ λ , | H i | ≤ λ , for i ∈ I ,and at least one member of C achieves the cardinality λ .We show that there exists a simple group S of cardinality λ that contains anisomorphic copy of each member of C and, for all H i , H i ′ in C with | H i ′ | = λ ,is generated by the copies of H i and H i ′ in S .This generalizes a result of Paul E. Schupp (moreover, our proof follows thesame approach based on small cancelation). In the countable case, we partiallyrecover a much deeper embedding result of Alexander Yu. Ol ′ shanski˘ı. Background and results
In [Sch76] Schupp used small cancelation theory (construction of Adian-Rabintype) to prove, among other things, the following result.
Theorem S (Schupp [Sch76]) . Let G , H and K be nontrivial groups with | G | ≤| H ∗ K | and | K | ≥ . There exists a simple group S that contains an isomorphiccopy of G and is generated by isomorphic copies of H and K . Corollary S.
Let G be a countable group. For all p, q ∈ { , , . . . } ∪ {∞} with q ≥ , there exists a simple group S that contains an isomorphic copy of G and isgenerated by a pair of elements of order p and q , respectively. The simple group constructed by Schupp in Theorem S, in addition to being de-pendent on G , depends on H and K . Accordingly, the simple group in Corollary S,in addition to being dependent on G , depends on the pair ( p, q ).We will show that the argument used by Schupp can be adapted in such a waythat the same simple group can be used even if one considerably varies H and K in Theorem S and, consequently, the same simple group can be used independentlyof the pair ( p, q ) in Corollary S. Theorem A.
Let | I | ≥ and C = { H i | i ∈ I } be a countable family of countablenontrivial groups, at least one of which has at least 3 elements (the groups may beisomorphic for different values of the index).There exists a 2-generated simple group S that contains an isomorphic copy ofeach member of C and, for all H i , H j in C with | H j | ≥ , is generated by the copiesof H i and H j in S . Mathematics Subject Classification.
Key words and phrases. simple groups, embeddings, small cancelation.Partially supported by NSF grant DMS-0600975.
Corollary A.
Let G be a countable group.There exists a simple group S that contains an isomorphic copy of G and, forall p, q ∈ { , , . . . } ∪ {∞} with q ≥ , is generated by a pair of elements of order p and q , respectively.Moreover, if | G | ≥ , then, for every p ∈ { , , . . . } ∪ {∞} , the simple group S is generated by G and an element of order p . In this note, countable means finite or countably infinite. The countability lim-itations imposed in Theorem A are natural since every countable group containsonly countably many finitely generated subgroups. An extension of Theorem S inwhich countability assumptions are not used follows.
Theorem B.
Let λ be an infinite cardinal number and let C = { H i | i ∈ I } be afamily of nontrivial groups. Assume that ≤ | I | ≤ λ , | H i | ≤ λ , for i ∈ I , and atleast one member of C achieves the cardinality λ .There exists a simple group S of cardinality λ that contains an isomorphic copyof each member of C and, for all H i , H i ′ in C with | H i ′ | = λ , is generated by thecopies of H i and H i ′ in S . Corollary B.
For any group G with | G | ≥ , there exists a simple group S thatcontains an isomorphic copy of G and, for every p ∈ { , , . . . } ∪ {∞} , is generatedby G and a single element of order p . In the countable case, the embedding results of Schupp were eventually subsumedby the following result of Ol ′ shanski˘ı (this result also subsumes our Theorem A,but not Theorem B). Theorem O (Ol ′ shanski˘ı [Ol ′ . Let | I | ≥ and C = { H i | i ∈ I } be a countablefamily of countable nontrivial groups.There exists a 2-generated simple group S that contains an isomorphic copy ofeach member of C and, moreover, has the following properties (in what follows, thecopy of H i in S is denoted by H i ). (1) If i, j ∈ I , i = j , | H j | ≥ , then S is generated by H i and H j . (2) If i, j ∈ I , i = j , then H i ∩ H j = 1 . (3) Every element of finite order in S is conjugate to an element in H i , for some i ∈ I . (4) Every proper subgroup of S is either infinite cyclic, or infinite dihedral, or itconjugate of a subgroup of H i , for some i ∈ I . (5) If, for some i ∈ I , x ∈ H i , x = 1 , y H i , then either S is generated by { x, y } or both x and y are involutions, or both x and xy are involutions, (6) If i, j ∈ I , i = j , then H i ∩ H xj = 1 , for every element x in S (7) For every i ∈ I , H i is malnormal in S (for every x ∈ S \ H i , H i ∩ H xi = 1 ). Thus there is a natural trade off in our approach. We extend Theorem S ofSchupp (by adapting his approach using small cancelation theory) to arbitraryfamilies of groups in a way that, in the countable case, partially recovers Theorem Oof Ol’shanski˘ı. A modest gain is achieved by the fact that the taken approach allowsus to handle families of groups that are not necessarily countable. On the otherhand, in the countable case, we recover only a small subset of the conclusions thatare obtained by the more powerful (but also more onerous) graded diagram methodsintroduced by Olshanski˘ı.
MBEDDINGS INTO SIMPLE GROUPS 3 Proofs and additional comments
Proof of Theorem A.
Reindex the family C (if necessary) so that it is indexed byan initial segment I of the set of natural numbers N = { , , . . . } (including thepossibility I = N , if I is infinite). Moreover, in case the cyclic group C of order 2is a member of C set H = C and make sure that this is the only copy of C in C .For each i ∈ I , embed H i into a 2-generated simple group S i = h s i , t i i (this canbe done by Theorem S) and consider the free product F = A ∗ B ∗ ( ∗ i ∈ I S i ), where A = h a | a i = C , B = h b | b i = C .For each index i ∈ I define the words u i = ( ab ) (2 i +1) n + n ( ab − ) ( ab ) (2 i +1) n + n − ( ab − ) . . . ( ab − ) ( ab ) (2 i +1) n +1 s i ,v i = ( ab ) (2 i +2) n + n ( ab − ) ( ab ) (2 i +2) n + n − ( ab − ) . . . ( ab − ) ( ab ) (2 i +2) n +1 t i , where n is a positive integer to be specified at a later stage.Choose a nontrivial element h in H and, for each i >
0, choose a pair of distinctnontrivial elements h i and ¯ h i in H i . For each pair of indices i, j ∈ I with 0 ≤ i < j ,define the words w ( a,i,j ) = ( h i h j ) n ( h i ¯ h j ) ( h i h j ) n − ( h i ¯ h j ) . . . ( h i ¯ h j ) ( h i h j ) aw ( b,i,j ) = ( h i h j ) n ( h i ¯ h j ) ( h i h j ) n − ( h i ¯ h j ) . . . ( h i ¯ h j ) ( h i h j ) n +1 b. Let R be the set of words obtained by symmetrization (closure under inversionand conjugation; see Remark 1 for a precise definition) of the set of words R ′ = { w ( a,i,j ) , w ( b,i,j ) | i, j ∈ I, ≤ i < j } ∪ { u i , v i , ( h i a ) n , ( h i b ) n | i ∈ I } and let H = h F | R i .Choose n that is relatively prime to 6 and is sufficiently large to ensure thatthe set of words R satisfies the small cancelation condition C ′ (1 /
6) over the freeproduct F = A ∗ B ∗ ( ∗ i ∈ I S i ) (see Remark 1 for a definition of the small cancelationcondition over free products). It follows, by a result of Lyndon [Lyn66, Theorem IV](see [LS01, Section V.9] for an exposition), that all factors in the free product F are embedded in H = h F | R i .The u relators and the v relators ensure that H is generated by a and b . On theother hand, the w relators ensure that H is generated by H i and H j for any i, j ∈ I with 0 ≤ i < j .Let M be a maximal normal subgroup of H and let S = H/M . The group S is simple by the maximality of M . We claim that all the factors S i , i ∈ I , arestill embedded in S . The factor S i , being simple, either intersects M trivially or iscontained in M . In the former case, the factor S i is still embedded in S = H/M .The latter case implies that h i = 1 in S . Because of the relators ( h i a ) n and ( h i b ) n ,it follows that a n = b n = 1 in S . However, n is chosen to be relatively prime to 6.Thus a = b = 1 in S , which means that S is trivial, a contradiction.This completes the proof. (cid:3) We note here the crucial role of the embeddings H i ֒ → S i in the course of theproof. On one hand, the number of generators needed for each factor in ∗ i ∈ I S i isuniformized. This is notationally convenient, but not crucial. More significant isthe simplicity of the factors S i , which, helped by the relators ( h i a ) n and ( h i b ) n ,“protects” the embedded subgroups H i from “crashing” when M is factored outfrom H . ZORAN ˇSUNI´C
Proof of Corollary A.
Apply Theorem A to C = { H i | i ≥ } , where H = C , H = G , and H i − = H i − = C i , for i ≥ C m denotes the cyclic group of order m ). (cid:3) Proof of Theorem B.
Let J be an indexing set of cardinality λ . For each i ∈ I ,embed H i into a simple group S i = h{ s i,j | j ∈ J }i . The cardinality of thesimple group S and the generating system of S i can be chosen to be equal to λ = | J | by Theorem S. Consider the free product F = A ∗ B ∗ ( ∗ i ∈ I S i ), where A = h a | a i = C , B = ∗ j ∈ J h b j | b j i = ∗ j ∈ J C .Let α : I × J → J be an injective map (such a map exists since | I | ≤ | J | and | J | is infinite).For each pair ( i, j ) ∈ I × J , define the word u i,j = ( ab α ( i,j ) ) n ( ab − α ( i,j ) ) ( ab α ( i,j ) ) n − ( ab − α ( i,j ) ) . . . ( ab − α ( i,j ) ) ( ab α ( i,j ) ) s i,j , where n is a positive integer to be specified at a later stage.For each i ∈ I , choose a nontrivial element h i in H i . For each i ′ ∈ I such that | H i ′ | = λ , choose a nontrivial element ¯ h i ′ in H i ′ different from h i ′ and distinctnontrivial elements h i ′ ,j , j ∈ J , ¯ h i ′ ,j , j ∈ J , in H i ′ that are also different from h i ′ and ¯ h i ′ . Let L ⊆ I × I be a set of pairs such that, for each pair of indices i, i ′ ∈ I such that | H i | < | H i ′ | = λ , the ordered pair ( i, i ′ ) is in L , and, for eachpair of indices i, i ′ ∈ I such that i = i ′ and | H i | = | H i ′ | = λ , exactly one of theordered pairs ( i, i ′ ) and ( i ′ , i ) is in L (if more than one member of C has cardinality λ , then there are many possible choices for L and we select one; the set L must benonempty because | I | ≥ i, i ′ ) in L , define the words w ( a,i,i ′ ) = ( h i h i ′ ) n ( h i ¯ h i ′ ) ( h i h i ′ ) n − ( h i ¯ h i ′ ) . . . ( h i ¯ h i ′ ) ( h i h i ′ ) aw ( b j ,i,i ′ ) = ( h i h i ′ ,j ) n ( h i ¯ h i ′ ,j ) ( h i h i ′ ,j ) n − ( h i ¯ h i ′ ,j ) . . . ( h i ¯ h i ′ ,j ) ( h i h i ′ ,j ) b j , j ∈ J. For all i ∈ I and j ∈ J also define the words( h i a ) n , ( h i b j ) n . The group H = F/R is defined as before ( R is obtained by symmetrization ofthe set of all words defined so far and n is chosen to be relatively prime to 6 and tobe sufficiently large to yield the cancelation condition C ′ (1 /
6) over the free product F = A ∗ B ∗ ( ∗ i ∈ I S i )). The group S = H/M , where M is a maximal normalsubgroup of H satisfies the required conditions. (cid:3) Remark 1.
We recall here the definition of the small cancelation property C ′ (1 / n in the proof of Theorem A that ensuresthat this condition is satisfied.Let G = ∗ i ∈ I G i be a free product of a nonempty family of nontrivial groups G i , i ∈ I (here I is an arbitrary nonempty indexing set). The free product G isgenerated by the set Σ = ∪ i ∈ I G i \ { } of nontrivial elements in the (disjoint) unionof the factors of G . A word g g . . . g k over Σ is reduced if, for ℓ = 1 , . . . , k − g ℓ and g ℓ +1 come from a different factor of G . Every element g in G can be representedby a unique reduced word over Σ, called the normal form of g . By definition, thelength of an element in G is equal to the length of its normal form. A reduced word g g . . . g k over Σ is weakly cyclically reduced if k ≤ g k g = 1 (thus g k and g may come from the same factor of G , but may not cancel). Let u = u . . . u k and v = v . . . v m be two reduced words over Σ. If k = 0 or m = 0 or u k v = 1, wesay that the product uv is semi-reduced (thus u k and v may come from the same MBEDDINGS INTO SIMPLE GROUPS 5 factor of G , but may not cancel). A set R of words over Σ is called symmetrizedif it consists of weakly cyclically reduced words, it is closed for inversion, and, forevery r in R , all weakly cyclically reduced words over Σ representing conjugates of r are in R . A nonempty reduced word p over Σ is a piece in the symmetrized set R if there exist two reduced words q and q and two distinct words r and r in R such that r = pq , r = pq in G and the products pq and pq are semi-reduced.In this case we say that p is a piece in r (and in r ). Note that p does not haveto be a subword of r to be a piece in it. A symmetrized set R of words over Σsatisfies the small cancelation property C ′ (1 /
6) over the free product G if, everyword in R has length greater than 6 and, for every piece p , every reduced word q ,and every word r in R such that r = pq in G and the product pq is semi-reduced,the inequality | p | < | r | holds.We now go back to our concrete setup from the proof of Theorem A. The lengthsof the words in the set of relators R ′ are | u i | = 4 n i + 3 n + 3 n − , | w ( a,i,j ) | = n + 3 n − , | ( h i a ) n | = 2 n, | v i | = 4 n i + 5 n + 3 n − , | w ( b,i,j ) | = 3 n + 3 n − , | ( h i b ) n | = 2 n. The reduced word p = ( h i h j ) n ( h i ¯ h j )( h i h j ) n of length 4 n + 2 is a piece in w ( a,i,j ) ,since it is a subword of w ( b,i,j ) (and whence a prefix in a cyclic conjugate of w ( b,i,j ) )and since w ( a,i,j ) can be written as a semi-reduced product pq (for an appropriatelychosen reduced word q with first letter h − j ¯ h j ). It can be easily verified that w ( a,i,j ) does not have a piece longer than p (although it has other pieces of the same length).Thus n needs to be selected in such a way that4 n + 2 n + 3 n − < . This is true for any n ≥
22, but since we require n to be relatively prime to 6, thesmallest good choice is n = 23.We may now fix n = 23, consider all other pieces of words, and verify that the C ′ (1 /
6) condition is satisfied.For instance, the word ( ab ) (2 i +1) n + n ( ab − )( ab ) (2 i +1) n + n of length 8 ni + 8 n + 2is a piece of u i (and this word does not have any longer pieces). Thus we need toverify that 8 ni + 8 n + 24 n i + 3 n + 3 n − < , for all i ≥
0. Think of the fraction on the left as a function of i . Since 8 n (3 n +3 n − − (8 n + 2) · n <
0, this function is decreasing for i ≥
0, the maximum isachieved at i = 0, and its value is (8 n + 2) / (3 n + 3 n −
1) = 186 / < / n = 23. Remark 2.
Consider again the proof of Theorem A. We used the original work ofSchupp not only to model our approach, but also to embed each group H i into asimple 2-generated group S i (in order to protect H i in the quotient S = H/M ). Inturn, in his proof of Theorem S, Schupp uses embeddings of G , H , and K into count-able simple groups. At about the same time Schup proved his result, Goryushkinalso proved that every countable group can be embedded into a 2-generated simplegroup [Gor74]. Before the results of Schupp and Goryushkin, it was known from thework of P. Hall that every countable group can be embedded into a 3-generated sim-ple group [Hal74, Theorem C2]. However, both Hall and Goryushkin also base their ZORAN ˇSUNI´C proofs on the existence of embeddings of countable groups into countable simplegroups. Thus to get back on some firm footing one could perhaps go back directlyto the classical embedding results of Higman, Neumann and Neuman. Namelythey prove [HNN49] that every countable group can be embedded into a countablegroup in which any two elements that have the same order are conjugate. As acorollary, every countable group can be embedded into a countable simple divisiblegroup (see [LS01, Theorem IV.3.4] for an exposition). Of course, by using suchembeddings directly in the course of the proof of Theorem A, we could skip over alayer in the construction at the cost of a mild notational difficulty (one would haveto deal with countably many countable generating sets).
Acknowledgments.
The author would like to thank Centre Interfacultaire Bernoulliat EPF-Lausanne for the support, the staff members for the hospitality during hisstay in May 2007, Goulnara Arzhantseva and Alain Valette for their kind invitationto participate in the program, and the referee for his/her rather useful remarks.
References [Gor74] A. P. Gorjuˇskin. Imbedding of countable groups in 2-generator simple groups.
Mat.Zametki , 16:231–235, 1974.[Hal74] P. Hall. On the embedding of a group in a join of given groups.
J. Austral. Math. Soc. ,17:434–495, 1974. Collection of articles dedicated to the memory of Hanna Neumann,VIII.[HNN49] Graham Higman, B. H. Neumann, and Hanna Neumann. Embedding theorems forgroups.
J. London Math. Soc. , 24:247–254, 1949.[LS01] Roger C. Lyndon and Paul E. Schupp.
Combinatorial group theory . Classics in Mathe-matics. Springer-Verlag, Berlin, 2001.[Lyn66] Roger C. Lyndon. On Dehn’s algorithm.
Math. Ann. , 166:208–228, 1966.[Ol ′
89] A. Yu. Ol ′ shanski˘ı. Efficient embeddings of countable groups. Vestnik Moskov. Univ.Ser. I Mat. Mekh. , (2):28–34, 105, 1989.[Sch76] Paul E. Schupp. Embeddings into simple groups.
J. London Math. Soc. (2) , 13(1):90–94,1976.
Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA
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