In Theorem 1.2 we may put L = G, Hi = ~k_z(G) (l = 0,1 .... , k) and Hi = 1 (l >/~).We find that for 0Further, for any …
In Theorem 1.2 we may put L = G, Hi = ~k_z(G) (l = 0,1 .... , k) and Hi = 1 (l >/~).We find that for 0Further, for any group G we denote by G >~ G' >~ G" ~ ... >~ G (t-1) >1 G ~f~ >1
Burnside ((1), p. 241) has proved the following theorem: If G is a non-metabelian p-group, then the centre of the derived group of G cannot be cyclic. In particular, a …
Burnside ((1), p. 241) has proved the following theorem: If G is a non-metabelian p-group, then the centre of the derived group of G cannot be cyclic. In particular, a non-Abelian group of order p 3 cannot be the derived group of a p-group .
1. Statement of the theorem. The aim of the present note is to investigate possible generalizations of the well-known fact that if a is a nonidentity element of a finitely-generated …
1. Statement of the theorem. The aim of the present note is to investigate possible generalizations of the well-known fact that if a is a nonidentity element of a finitely-generated nilpotent group G, there exists an epimorphism 4 of G onto a finite group such that acu P 1. The generalization that we consider is the following. Let G be a finitely-generated nilpotent group, and let w(xi, * * *, Xn; a,, * * *, am) be a word in variables x1, * * *, xn and elements a1, * * *, am of G. If w = 1 has no solution in G, does there exist an epimorphism 4 of G onto a finite group H such that w(x1, * , Xn; a14, * * * , am+) =1 has no solution in H? The answer in general is in the negative, as is shown by a counterexample constructed below. However, we shall prove, in answer to a question posed by A. W. Mostowski, that the answer is in the affirmative if w = x-laxb-1. Our aim then is to prove the following.
Let G denote a group of order a power of the prime p , and let G ′ be the derived group of G . The lower central series of …
Let G denote a group of order a power of the prime p , and let G ′ be the derived group of G . The lower central series of G will be written For any subgroup H of G we denote by P ( H ) the subgroup of H generated by all elements x p as x runs through H , and by Φ( H ) the Frattini subgroup of H . We write ( H :Φ( H )) = p d(H) ; thus d ( H ) is the minimal number of generators of H .
Magnus [4] proved the following theorem. Suppose that F is free group and that X is a basis of F . Let R be a normal subgroup of F and …
Magnus [4] proved the following theorem. Suppose that F is free group and that X is a basis of F . Let R be a normal subgroup of F and write G = F / R . Then there is a monomorphism of F / R ′ in which ; here the t x are independent parameters permutable with all elements of G . Later investigations [1, 3] have shown what elements can appear in the south-west corner of these 2 × 2 matrices. In this form the theorem subsequently reappeared in proofs of the cup-product reduction theorem of Eilenberg and MacLane (cf. [7, 8]). In this note a direct group-theoretical proof of the theorems will be given.
Burnside[1] considered possible restrictions on the derived group G ′ of p -group G and showed that if G ′ is non-Abelian, the centre Z ( G ′) of G …
Burnside[1] considered possible restrictions on the derived group G ′ of p -group G and showed that if G ′ is non-Abelian, the centre Z ( G ′) of G ′ is not cyclic. This implies that | G ′: G ″| ≥ p 3 . Many other restrictions on G ′ are to be found in Hall's famous paper [2], but in 1954 Hall proved that if p is odd and | G ′: G ″| = p 3 , then | G ′| ≤ p . So far as I know, no proof of this is to be found in the literature, but it follows from the lemma below. Our concern here is with the case p = 2, and we shall prove the following.
If H, K are subgroups of a group G , then HK is a subgroup of G if and only if HK = KH . This condition certainly holds if …
If H, K are subgroups of a group G , then HK is a subgroup of G if and only if HK = KH . This condition certainly holds if H ≤ N G ( K ) or K ≤ N G ( H ). But the majority of groups can also be expressed as HK , where neither H nor K is normal. In this paper we consider groups G for which no subgroup G 1 can be expressed as the product of non-normal subgroups of G 1 . Such a group is said to be equilibrated. Thus G is equilibrated if and only if either H ≤ N G ( K ) or K ≤ N G ( H ) whenever H, K and HK are subgroups of G .
If H, K are subgroups of a group G , then HK is a subgroup of G if and only if HK = KH . This condition certainly holds if …
If H, K are subgroups of a group G , then HK is a subgroup of G if and only if HK = KH . This condition certainly holds if H ≤ N G ( K ) or K ≤ N G ( H ). But the majority of groups can also be expressed as HK , where neither H nor K is normal. In this paper we consider groups G for which no subgroup G 1 can be expressed as the product of non-normal subgroups of G 1 . Such a group is said to be equilibrated. Thus G is equilibrated if and only if either H ≤ N G ( K ) or K ≤ N G ( H ) whenever H, K and HK are subgroups of G .
Burnside[1] considered possible restrictions on the derived group G ′ of p -group G and showed that if G ′ is non-Abelian, the centre Z ( G ′) of G …
Burnside[1] considered possible restrictions on the derived group G ′ of p -group G and showed that if G ′ is non-Abelian, the centre Z ( G ′) of G ′ is not cyclic. This implies that | G ′: G ″| ≥ p 3 . Many other restrictions on G ′ are to be found in Hall's famous paper [2], but in 1954 Hall proved that if p is odd and | G ′: G ″| = p 3 , then | G ′| ≤ p . So far as I know, no proof of this is to be found in the literature, but it follows from the lemma below. Our concern here is with the case p = 2, and we shall prove the following.
Magnus [4] proved the following theorem. Suppose that F is free group and that X is a basis of F . Let R be a normal subgroup of F and …
Magnus [4] proved the following theorem. Suppose that F is free group and that X is a basis of F . Let R be a normal subgroup of F and write G = F / R . Then there is a monomorphism of F / R ′ in which ; here the t x are independent parameters permutable with all elements of G . Later investigations [1, 3] have shown what elements can appear in the south-west corner of these 2 × 2 matrices. In this form the theorem subsequently reappeared in proofs of the cup-product reduction theorem of Eilenberg and MacLane (cf. [7, 8]). In this note a direct group-theoretical proof of the theorems will be given.
1. Statement of the theorem. The aim of the present note is to investigate possible generalizations of the well-known fact that if a is a nonidentity element of a finitely-generated …
1. Statement of the theorem. The aim of the present note is to investigate possible generalizations of the well-known fact that if a is a nonidentity element of a finitely-generated nilpotent group G, there exists an epimorphism 4 of G onto a finite group such that acu P 1. The generalization that we consider is the following. Let G be a finitely-generated nilpotent group, and let w(xi, * * *, Xn; a,, * * *, am) be a word in variables x1, * * *, xn and elements a1, * * *, am of G. If w = 1 has no solution in G, does there exist an epimorphism 4 of G onto a finite group H such that w(x1, * , Xn; a14, * * * , am+) =1 has no solution in H? The answer in general is in the negative, as is shown by a counterexample constructed below. However, we shall prove, in answer to a question posed by A. W. Mostowski, that the answer is in the affirmative if w = x-laxb-1. Our aim then is to prove the following.
Let G denote a group of order a power of the prime p , and let G ′ be the derived group of G . The lower central series of …
Let G denote a group of order a power of the prime p , and let G ′ be the derived group of G . The lower central series of G will be written For any subgroup H of G we denote by P ( H ) the subgroup of H generated by all elements x p as x runs through H , and by Φ( H ) the Frattini subgroup of H . We write ( H :Φ( H )) = p d(H) ; thus d ( H ) is the minimal number of generators of H .
In Theorem 1.2 we may put L = G, Hi = ~k_z(G) (l = 0,1 .... , k) and Hi = 1 (l >/~).We find that for 0Further, for any …
In Theorem 1.2 we may put L = G, Hi = ~k_z(G) (l = 0,1 .... , k) and Hi = 1 (l >/~).We find that for 0Further, for any group G we denote by G >~ G' >~ G" ~ ... >~ G (t-1) >1 G ~f~ >1
Burnside ((1), p. 241) has proved the following theorem: If G is a non-metabelian p-group, then the centre of the derived group of G cannot be cyclic. In particular, a …
Burnside ((1), p. 241) has proved the following theorem: If G is a non-metabelian p-group, then the centre of the derived group of G cannot be cyclic. In particular, a non-Abelian group of order p 3 cannot be the derived group of a p-group .
Burnside ((1), p. 241) has proved the following theorem: If G is a non-metabelian p-group, then the centre of the derived group of G cannot be cyclic. In particular, a …
Burnside ((1), p. 241) has proved the following theorem: If G is a non-metabelian p-group, then the centre of the derived group of G cannot be cyclic. In particular, a non-Abelian group of order p 3 cannot be the derived group of a p-group .
In Theorem 1.2 we may put L = G, Hi = ~k_z(G) (l = 0,1 .... , k) and Hi = 1 (l >/~).We find that for 0Further, for any …
In Theorem 1.2 we may put L = G, Hi = ~k_z(G) (l = 0,1 .... , k) and Hi = 1 (l >/~).We find that for 0Further, for any group G we denote by G >~ G' >~ G" ~ ... >~ G (t-1) >1 G ~f~ >1
1. If P is any property of groups, then we say that a group G is ‘locally P’ if every finitely generated subgroup of G satisfies P. In this paper …
1. If P is any property of groups, then we say that a group G is ‘locally P’ if every finitely generated subgroup of G satisfies P. In this paper we shall be chiefly concerned with the case when P is the property of being nilpotent, and will examine some properties of nilpotent groups which also hold for locally nilpotent groups. Examples of locally nilpotent groups are the locally finite p -groups (groups such that every finite subset is contained in a finite group of order a power of the prime p ); indeed, every periodic locally nilpotent group is the direct product of locally finite p -groups.
Abstract We present a direct combinatorial proof of the characterization of the degree of transivity of a finite permutation group in terms of the Bell numbers. Subject classification ( Amer. …
Abstract We present a direct combinatorial proof of the characterization of the degree of transivity of a finite permutation group in terms of the Bell numbers. Subject classification ( Amer. Math.Soc. ( MOS ) 1970 ): 20 B 20.
Notation and definitions 384 3. Statement of main theorem and corollaries 388 4. Proofs of corollaries 389 5. Preliminary lemmas 389 5.1.Inequalities and modules 389 5.2.7r-reducibility and fl,(®) 395 5.3.Groups …
Notation and definitions 384 3. Statement of main theorem and corollaries 388 4. Proofs of corollaries 389 5. Preliminary lemmas 389 5.1.Inequalities and modules 389 5.2.7r-reducibility and fl,(®) 395 5.3.Groups of symplectic type 397 5.4.^-groups, ^-solvability and F((&) 400 5.5.Groups of low order 404 5.6.2-groups, involutions and 2-length 409 5.7.Factorizations 423 5.8.Miscellaneous 426 6.A transitivity theorem 428 i 9 68]
Let G denote a group of order a power of the prime p , and let G ′ be the derived group of G . The lower central series of …
Let G denote a group of order a power of the prime p , and let G ′ be the derived group of G . The lower central series of G will be written For any subgroup H of G we denote by P ( H ) the subgroup of H generated by all elements x p as x runs through H , and by Φ( H ) the Frattini subgroup of H . We write ( H :Φ( H )) = p d(H) ; thus d ( H ) is the minimal number of generators of H .
Article Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen. was published on January 1, 1911 in the journal Journal für die reine und angewandte Mathematik …
Article Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen. was published on January 1, 1911 in the journal Journal für die reine und angewandte Mathematik (volume 1911, issue 139).
INTRODUCTION.Le présent Mémoire est consacré a l'étude des p-groupes de Sylow des groupes symétriques dont le degré est une puissance p 7 " d'un nombre premier p. Ces groupes, qui …
INTRODUCTION.Le présent Mémoire est consacré a l'étude des p-groupes de Sylow des groupes symétriques dont le degré est une puissance p 7 " d'un nombre premier p. Ces groupes, qui seront désignés par lH/n, jouent, relativement aux p-groupes, un rôle analogue à celui que les groupes symétriques jouent relativement aux groupes finis.Le fait de considérer seulement des groupes symétriques de degré ^m n'est pas une restriction artificielle.Il est facile de montrer que les p-groupes de Sylow d'un groupe symétrique quelconque sont des produits directs de tels groupes |ll^.De façon précise, si le nombre entier n s'écrit dans le système de numération de base?:n == ciQ + a^p + a,? 2 -4-... 4-âmp 11 ûn p-groupe de Sylow du groupe symétrique de degré n est isomorphe au produit direct îi?xy?x...x^r.On peut représenter les groupes |ll^par des systèmes, que j'appelle tableaux, de m polynômes, à m-i variables, dont les coefficients sont des classes d'entiers, modp (ou des éléments d'un champ de Galois de p éléments).Entre ces systèmes existe une loi de composition, définie par des opérations 2/i0 L. KALOUJNINË.rationnelles relativement simples.On peut ainsi traduire tout problème sur les groupes T^m 6n un problème sur les polynômes, dans un champ de Galois.Il est vrai que, dans la plupart des cas, on aboutit ainsi à d^s problèmes qui sont loin d'être résolus.Cependant il est des cas pour lesquels on obtient une réponse satisfaisante et quasi définitive.C'est ainsi qu^on peut déterminer tous les sous-groupes caractéristiques d'un groupe ^1,», et les caractériser par des systèmes de 772 entiers satisfaisant à certaines conditions.Les résultats obtenus sont formulés dans les Chapitres IV et V, sous forme d'un certain nombre de théorèmes.Les démonstrations en sont basées sur quelques lemmes simples et sur quelques inégalités qui sont établis aux Chapitres II et III; le Chapitre 1 indiquant le principe de la représentation des groupes par des tableaux de.polynômes.Si ces inégalités pouvaient être précisées, elles permettraient sans doute de trouver d'autres propriétés intéressantes des groupes |î^ et de leurs sous-groupes.Cependant le problème de la détermination de tous les sous-groupes des groupes |P/n et de leur description précise semble exiger des moyens qui dépassent de loin nos connaissances actuelles.Il peut néanmoins être traité complètement dans le-cas des groupes IL.C'est c*e que j'ai fait dans le Chapitre VI.Une grande partie des résultats que j'ai ainsi obtenus semble avoir été trouvée déjà par d'autres méthodes.J'ai pensé qu'il y avait quelque intérêt à les reprendre avec l'algorithme du présent travail.On peut voir mieux ainsi le mécanisme du calcul, et il n'est pas impossible de penser que leur étude pourra servir à une étude ultérieure des groupes ^3.Dans un autre Mémoire qui sera publié ultérieurement, j^étudie quelques généralisations des idées dont je me suis servi dans le présent travail, en étudiant les groupes |1^ des tableaux formés par une suite infinie dénombrable de polynômes à coefficients dans des champs de Galois.Le présent Mémoire et sa généralisation ont été présentés à l'Université de Paris comme Thèse de Doctorat.J'exprime toute ma gratitude à MM.É. Cartan, G. Darmois et A. Châtelet, qui ont bien voulu être membres du Jury.Je tiens à remercier plus spécialement M. Châtelet pour le vif intérêt qu^il a pris au présent travail et les conseils qu'il m'a prodigués, pour son développement et pour sa rédaction.J'adresse encore mes remereîments à M. H. Cartan pour l'intérêt qu'il a pris à mes recherches et pour ses encouragements, et à M. Krasner dont les critiques et les conseils m'ont été précieux et m'ont permis d^améliorer mon exposé.Je suis également très reconnaissant à MM.É. Cartan et J. Hadamard qui, à plusieurs reprises, ont présenté mes Notes sur ces sujets à PAcadémie des Sciences, et plus spécialement à M. Paul Montel, qui n'a pas hésité à accueillir et faire publier une de mes Noies, pendant mon internement par la Gestapo, au camp de Compiègne; en acceptant ce travail dans les Annales de l ) École Normale Les groupes abéliens de type (p, p, . .., p) sont appelés élémentaires.Le groupesymétrique', de degré n (et d'ordre // ! ) est désigné par JS/,.Le p-groupe de Sylow de JS^,,,, défini à un isomorphisme près, est désigné par | 11^.,0n utilise les signes ordinaires de la théorie des ensembles :C inclusion stricte; <^ inégalité stricte; Ç inclusion large, ^inégalité large. CHAPITRE I.LKS TABLEAUX DE RANG m.Considérons un ensemble de T dep'" éléments et le groupe des permutations J? ,n de T^ c'est-à-dire le groupe symétrique de degré p'".La contribution àep dans son ordre p^! est p^'"" 1 ^'"-24 -M ; c'est l'ordre commun de ses p-groupes de Sylow, ^m (tous isomorphes entre eux).Étant donné un p-groupe quelconque (^, il existe une représentation fidèle de (S) comme groupe de permutations dont le degré est une puissance p 7 ".Alors, en raison des théorèmes de Sylow, dans tout 1|1/^ il existe un sous-•24 2 L. KALOUJNINE.
Reductions to local subgroups and sections Factorizations for $p=2$ The general situation Appendix A1. Proof of Theorem A Appendix A2. Corrections and additions to GL Bibliography.
Reductions to local subgroups and sections Factorizations for $p=2$ The general situation Appendix A1. Proof of Theorem A Appendix A2. Corrections and additions to GL Bibliography.
Among the groups with minimum condition one meets, often quite unexpectedly, groups which satisfy one of the following two conditions: (a) the commutator subgroup is finite; (b) there exists an …
Among the groups with minimum condition one meets, often quite unexpectedly, groups which satisfy one of the following two conditions: (a) the commutator subgroup is finite; (b) there exists an abelian subgroup of finite index. It is our objective in this investigation to give various characterizations of these two classes of groups some of which have very little obvious connection with either the minimum condition or properties (a) and (b). Our principal result, stated in ?0, contains characterizations of the groups with minimum condition and finite commutator subgroup; and the proof of this theorem is effected in ?1 to ?5. In ?6 we specialize this result to show that torsion groups with finite automorphism groups are finite. In ?7 we enunciate a characterization of the groups with minimum condition possessing abelian subgroups of finite index; and the proof of this proposition will be effected in ?8 to ?11. These results are used in ?12 to show that the p-Sylow subgroups of these groups are conjugate. 0. In this section we enunciate and discuss our principal
The Word Problem is that of determining when two elements of F (or two 'words') represent the same element of G, or, equivalently, when a given element W of F …
The Word Problem is that of determining when two elements of F (or two 'words') represent the same element of G, or, equivalently, when a given element W of F lies in R, and so can be written in the form (1). Magnus has solved this problem for the case n = 1, of a single defining relation.3 A complementary problem is that of uniqueness for expressions of the form (1), or more simply that of determining all identities among the relations R,, * **, Rn Xof the form
Journal Article ON p-GROUPS OF MAXIMAL CLASS I Get access C. R. LEEDHAM-GREEN, C. R. LEEDHAM-GREEN Department of Pure Mathematics, Queen Mary CollegeLondon E1 4NS Search for other works by …
Journal Article ON p-GROUPS OF MAXIMAL CLASS I Get access C. R. LEEDHAM-GREEN, C. R. LEEDHAM-GREEN Department of Pure Mathematics, Queen Mary CollegeLondon E1 4NS Search for other works by this author on: Oxford Academic Google Scholar SUSAN MCKAY SUSAN MCKAY Department of Pure Mathematics, Queen Mary CollegeLondon E1 4NS Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume 27, Issue 3, September 1976, Pages 297–311, https://doi.org/10.1093/qmath/27.3.297 Published: 01 September 1976 Article history Received: 15 July 1975 Published: 01 September 1976
A question of L.G. Kovács, Joachim Neubüser, B.H. Neumann ( J. Austral. Math. Soc. 12 (1971), 287–300) on the existence of ‘secretive’ prime-power groups of large rank is settled affirmatively …
A question of L.G. Kovács, Joachim Neubüser, B.H. Neumann ( J. Austral. Math. Soc. 12 (1971), 287–300) on the existence of ‘secretive’ prime-power groups of large rank is settled affirmatively by proving the following result: given a prime p and integer d ≥ 2, there exists a finite p –group P with cyclic centre and minimal number of generators d and having the property that every element not in its Frattini subgroup has a non-trivial power in its centre.