A note on Engel groups and local nilpotence

Let n be a positive integer. We say that a group G is an ( n + 1 2 ) {(n+\frac{1}{2})} -Engel group if it satisfies the law [ x , y n , x ] = 1 {[x,{}_{n}y,x]=1} . The variety of ( n + 1 2 ) {(n+\frac{1}{2})} -Engel groups lies between the varieties of n-Engel groups and ( n + 1 ) {(n+1)} -Engel groups. In this paper, we study these groups, and in particular, we prove that all ( 4 + 1 2 ) {(4+\frac{1}{2})} -Engel { 2 , 3 } {\{2,3\}} -groups are locally nilpotent. We also show that if G is a ( 4 + 1 2 ) {(4+\frac{1}{2})} -Engel p-group, where p ≥ 5 {p\geq 5} is a prime, then G p {G^{p}} is locally nilpotent.

Let G be a group and let us write [x, y] = x−1y−1xy. We shall say that G is an Engel group, or that it satisfies the Engel condition, if to any pair of elements x, y in G there can be assigned an integer k (which is allowed to depend on x, y) such that

In this paper, by considering the notions of polygroup and Engel group, we introduce the concept of Engel fuzzy subpolygroups. In this regard, by a normal Engel fuzzy subpolygroup [Formula: see text] of [Formula: see text] and [Formula: see text], the fundamental relation on a given polygroup [Formula: see text], we construct an Engel fuzzy subgroup [Formula: see text]. We obtain a necessary and sufficient condition between Engel fuzzy subpolygroups and the Engel group [Formula: see text]/[Formula: see text], the group of equivalence classes derived from a fuzzy subpolygroup of [Formula: see text]. Finally, by using these results, we get Zorn’s lemma, in the Engel fuzzy subpolygroups.

In 1992, Wilson and Zelmanov proved that a profinite Engel group is locally nilpotent. Here we prove the stronger result that every compact Engel group is locally nilpotent.

Let A A be a Noetherian Cohen-Macaulay domain, b b , c 1 c_1 , … \dots , c g c_g an A A -sequence, J J = ( b , c 1 , … , c g ) A (b,c_1,\dots ,c_g)A , and B B = A [ J / b ] A[J/b] . Then B B is Cohen-Macaulay, there is a natural one-to-one correspondence between the sets Ass B ⁡ ( B / b B ) \operatorname {Ass}_B(B/bB) and Ass A ⁡ ( A / J ) \operatorname {Ass}_A(A/J) , and each q q ∈ \in Ass A ⁡ ( A / J ) \operatorname {Ass}_A(A/J) has height g + 1 g+1 . If B B does not have unique factorization, then some height-one prime ideals P P of B B are not principal. These primes are identified in terms of J J and P ∩ A P \cap A , and we consider the question of how far from principal they can be. If A A is integrally closed, necessary and sufficient conditions are given for B B to be integrally closed, and sufficient conditions are given for B B to be a UFD or a Krull domain whose class group is torsion, finite, or finite cyclic. It is shown that if P P is a height-one prime ideal of B B , then P ∩ A P \cap A also has height one if and only if b b ∉ \notin P P and thus P ∩ A P \cap A has height one for all but finitely many of the height-one primes P P of B B . If A A has unique factorization, a description is given of whether or not such a prime P P is a principal prime ideal, or has a principal primary ideal, in terms of properties of P ∩ A P \cap A . A similar description is also given for the height-one prime ideals P P of B B with P ∩ A P \cap A of height greater than one, if the prime factors of b b satisfy a mild condition. If A A is a UFD and b b is a power of a prime element, then B B is a Krull domain with torsion class group if and only if J J is primary and integrally closed, and if this holds, then B B has finite cyclic class group. Also, if J J is not primary, then for each height-one prime ideal p p contained in at least one, but not all, prime divisors of J J , it holds that the height-one prime p A [ 1 / b ] ∩ B pA[1/b] \cap B has no principal primary ideals. This applies in particular to the Rees ring R {\mathbf R} = = A [ 1 / t , t J ] A[1/t, tJ] . As an application of these results, it is shown how to construct for any finitely generated abelian group G G , a monoidal transform B B = A [ J / b ] A[J/b] such that A A is a UFD, B B is Cohen-Macaulay and integrally closed, and G G ≅ \cong Cl ⁡ ( B ) \operatorname {Cl}(B) , the divisor class group of B B .

Let p p be an odd prime. Let P P be a finite p p -group of class 2 2 and exponent p p , whose commutator quotient P / [ P , P ] P/[P,P] is of order p n p^n . We define two parameters for P P related to central decompositions. The first parameter, κ ( P ) \kappa (P) , is the smallest integer s s for the existence of a subgroup S S of P P satisfying (1) S ∩ [ P , P ] = [ S , S ] S\cap [P,P]=[S,S] , (2) | S / [ S , S ] | = p n − s |S/[S,S]|=p^{n-s} , and (3) S S is centrally decomposable. The second parameter, λ ( P ) \lambda (P) , is the smallest integer s s for the existence of a central subgroup N N of order p s p^s , such that P / N P/N is centrally decomposable. While defined in purely group-theoretic terms, these two parameters generalise, respectively, the vertex and edge connectivities of graphs: For a simple undirected graph G G , through the classical procedures of Baer (Trans. Amer. Math. Soc., 1938), Tutte (J. Lond. Math. Soc., 1947) and Lovász (B. Braz. Math. Soc., 1989), there is a p p -group P G P_G of class 2 2 and exponent p p that is naturally associated with G G . Our main results show that the vertex connectivity κ ( G ) \kappa (G) equals κ ( P G ) \kappa (P_G) , and the edge connectivity λ ( G ) \lambda (G) equals λ ( P G ) \lambda (P_G) . We also discuss the relation between κ ( P ) \kappa (P) and λ ( P ) \lambda (P) for a general p p -group P P of class 2 2 and exponent p p , as well as the computational aspects of these parameters. In particular, our main results imply that the p p -group central decomposition algorithm of Wilson (J. Algebra and J. of Group Theory, 2009) can be used to solve the graph connectivity problem.

Major progress in group theory occurred in the nineteenth century but the evolution of group theory began already in the latter part of 18th century. Some characteristic features of 19th century mathematics which had crucial impact on this evolution are concern for rigor and abstraction and the view that mathematics is a human activity without necessarily referring to physical situations. In 1770, Joseph Louis Lagrange (1736-1813) wrote his seminal memoir Reflections on the solution of algebraic equations. He considered ‘abstract’ questions like whether every equation has a root and, if so, how many were real/complex/positive/negative? The problem of algebraically solving 5th degree equations was a major preoccupation (right from the 17th century) to which Lagrange lent his major efforts in this paper. His beautiful idea (now going under the name of Lagrange’s resolvent) is to ‘reduce’ a general equation to auxiliary (resolvent) equations which have one degree less. Later, the theory of finite abelian groups evolved from Carl Friedrich Gauss’s famous “Disquisitiones Arithmeticae”. Gauss (1777-1855) established many of the important properties though he did not use the terminology of group theory. In his work, finite abelian groups4 Math Unlimitedappeared in different forms like the additive group Zn of integers modulo n, the multiplicative group Z∗n of integers modulo n relatively prime to n, the group of equivalence classes of binary quadratic forms, and the group of n-th roots of unity. In 1872, Felix Klein delivered a famous lecture A Comparative Review of Recent Researches in Geometry. The aim of his (so-called) Erlangen Program was the classification of geometry as the study of invariants under various groups of transformations. So, the groups appear here “geometrically” as groups of rigid motions, of similarities, or as the hyperbolic group etc. During the analysis of the connections between the different geometries, the focus was on the study of properties of figures invariant under transformations. Soon after, the focus shifted to a study of the transformations themselves. Thus the study of the geometric relations between figures got converted to the study of the associated transformations. In 1874, Sophus Lie introduced his “theory of continuous transformation groups” what we basically call Lie groups today. Poincaré and Klein began their work on the so-called automorphic functions and the groups associated with them around 1876. Automorphic functions are generalizations of the circular, hyperbolic, elliptic, and other functions of elementary analysis. They are functions of a complex variable z, analytic in some domain, and invariant under the group of transformations x′ = ax+b(a, b, c, d real or complex and ad − bc 0), or under some subgroup of this group. We end this introduction with the ancient problem of finding all those positive integers (called congruent numbers) which are areas of right-angled triangles with rational sides. To this date, the general result describing all of them precisely is not proved. However, the problem can be re-stated in terms of certain groups called elliptic curves and this helps in getting a hold on the problem and in obtaining several partial results. Indeed, if n is a positive integer, then the rational numbers x, y satisfying y2 = x3 − n2x form a group law which can be described geometrically. Then, n is a congruent number if and only if, this group is infinite!

Elementary abelian groups are finite groups in the form of A = ( ℤ / p ⁢ ℤ ) r {A=(\mathbb{Z}/p\mathbb{Z})^{r}} for a prime number p. For every integer ℓ > 1 {\ell>1} and r > 1 {r>1} , we prove a non-trivial upper bound on the ℓ {\ell} -torsion in class groups of every A-extension. Our results are pointwise and unconditional. This establishes the first case where for some Galois group G, the ℓ {\ell} -torsion in class groups are bounded non-trivially for every G-extension and every integer ℓ > 1 {\ell>1} . When r is large enough, the unconditional pointwise bound we obtain also breaks the previously best known bound shown by Ellenberg and Venkatesh under GRH.

The goal of this paper is to prove theorems that elucidate the Cohen–Lenstra–Martinet conjectures for the distributions of class groups of number fields, and further the understanding of their implications. We start by giving a simpler statement of the conjectures. We show that the probabilities that arise are inversely proportional to the number of automorphisms of structures slightly larger than the class groups. We find the moments of the Cohen–Lenstra–Martinet distributions and prove that the distributions are determined by their moments. In order to apply these conjectures to class groups of non-Galois fields, we prove a new theorem on the capitulation kernel (of ideal classes that become trivial in a larger field) to relate the class groups of non-Galois fields to the class groups of Galois fields. We then construct an integral model of the Hecke algebra of a finite group, show that it acts naturally on class groups of non-Galois fields, and prove that the Cohen–Lenstra–Martinet conjectures predict a distribution for class groups of non-Galois fields that involves the inverse of the number of automorphisms of the class group as a Hecke-module.

Let r be a positive rational integer, and G a finite group. A subgroup H of G is called a r minimal (r-maximal) subgroup if there exists a subgroup chain 1 = H o ~< H 1 ~< -.. ~<H, = H(H = Ho ~< Hx ~<'" ~< H, = G), where H i is maximal in Hi+ x, we call Han S-quasinormal subgroup of G, if HP = PH for any Sylow subgroup P of G. If H is S-quasinormal in G, we write H SQN G. A subgroup H of G is called a quasinormal subgroup if HK = KH for any subgroup K of G. We write H QN G when H is a quasinormal subgroup of G. G is said to be an SQN-r-group(QN-r-group, PN-r-group) if every r-minimal subgroup of G is S-quasinormal(quasinormal, normal) in G. Clearly, a PN-r-group is a ON-r-group, and a QN-r-group is an SQN-r-group. Gaschfitz and Ito proved that PNl -g roup G is solvable and its commutator subgroup G' isp-nilpotent for any odd prime p. Hence the Fitting length Fl(G) ~< 3 [5, th5.7 p436]. Buckley proved in [1] that PN-1group of odd order is supersolvable. Sastry and Deskins had studied in I'3] PN-r-group and QNr-group. For instance, they proved that QN-r-group (r = 1,2,3) is solvable and QN-l-group and QN-2-group have Fitting length at most 4, the Fitting length of QN-3-group is not more than 5. This paper will study the SQN-r-groups. In section 2, we explore the properties of S--quasinormal subgroup and prove that the normal Hall subgroups of an S-quasinormal subgroup of a finite group is also an S-quasinormal subgroup. In section 3, we prove that SQN-l-group is solvable and the nitpotent residual is p-nilpotent for any odd prime p, so the Fitting length is at most 3. In section 4, we prove that SQN-2-group and SQN-3-group are either supersolvable or isomorphic to the several classes groups given in section t. Moreover, the commutator subgroups are nilpotent, so the Fitting lengths are at most 2. In section 5, we show that SQN-4-group and SQN-5--group are either solvable and the Fitting lengths at most 3 or non-solvable with only one non-abelian principal factor ~r is isomorphic ~o the simple group PSL(2,p). Groups in this .paper are all of finite order. The author wishes to thank his supervisor Professor Duan Xuefu and-Xu Mingyiao for their encouragement and many instructions.

A finite group G is said to be a minimal non-[Formula: see text] group if G is not a group of class ≤ n whose proper subgroups are of class ≤ n. In this paper, we give a complete classification of p-groups H of odd order with d(H) = 2 and c(H) = 2. Based on the classification of H, minimal non-[Formula: see text]p-groups G are classified for p &gt; 3. If p &gt; 3, then we have G3 ≅ Cp or G3 ≅ Cp × Cp. In this paper, we deal with the case when G3 ≅ Cp. In another paper [A classification of finite p-groups whose proper subgroups are of class ≤ 2(II), accepted] we deal with the case when G3 ≅ Cp × Cp.

Let K be a field. The Schur subgroup, S(K), of the Brauer group, B(K), consists of all classes [△] in B(K) some representative of which is a simple component of one of the semi-simple group algebras, KG, where G is a finite group such that char K ∤ G. Yamada ([11], p. 46) has characterized S(K) for all finite extensions of the p-adic number field, Qp. If p is odd, [△] ∈ S(K) if and only ifwhere c is the tame ramification index of k/Qp, k the maximal cyclotomic subfield of K, and s = ((p – 1)/c, [K : k]). invp △ is the Hasse invariant. Yamada showed this by proving first that S(K) is the group of classes containing cyclotomic algebras and then determining the invariants of such algebras.

Let k be an algebraic number field (of finite degree) and K/k be a (finite) Galois extension with Galois group R. In connection with the class field theory, Weil [101 derived a certain canonical factor set a(o, r) of 9 in the idele-class group (K of K, which is uniquely determined by K/k in the sense of equivalence. For instance, a (mod R') -?f t r(o, A) (Ek k, essentially) (mod NK/k( RK)) is the Artin reciprocity law isomoprhism of 9/9' and Sk/NK/k( SK), where 9' is the commutator subgroup of R. A different approach to this (idele-class) factor set was given in [6]. Let, for each pair o, r, a(o, r) be a representative idele of a(o, r). Then (3a)(p, o, r) = a(o, r)a(po, r)-'a(p, o-r)a(p, oY)is, for each triple p, a, r, a principal idele, i.e. a non-zero element of K. We obtain thus a 3-cochain a(p, a, r) = (6a)(p, a, r) of 9 in the principal idele group PK of K, which is in fact a 3-cocycle. We saw, in [6], that this 3-cycocle a, or more precisely its cohomology class, attached in invariant fashion to the Galois extension K/k, has the order m/m', where m = (K: k) and m' is the least common multiple of the p-degrees of K/k, p running over all primes in k. It was shown further, in a joint paper [4] by Hochschild and the writer, that the cyclic group, of order m/m', generated by this 3-cohomology class is exactly the group of all 3-cohomology classes of 9 in PK with ceiling. This last group is in turn, according to EilenbergMacLane [1], nothing but the group of all Teichmtiller 3-cohomology classes for K/k, belonging to central simple algebras over K to which every element of ft can be extended (as automorphism). Now arises the problem to determine the exact algebra-class (though not unique) which possesses our a, attached to K/k, as its Teichmiiller-class. The present work is devoted to this problem and our solution is given in our Theorem in ?3. If a central simple algebra A over K, to which every automorphism of K/k can be extended, is given in a crossed-product form for instance, we can obtain its Teichmiiller-cocycle in a more or less explicit process involving the HilbertSpeiser theorem. On the other hand, in order to obtain our 3-cocycle a, we repeat the construction of the canonical idele-class factor set a given in [6], which is nothing but the global analogy of the local construction in Hochschild [3]1 and makes use of the idele interpretation of the principal genus theorem of Noether, but with more detailed analysis. Namely, we carry out all the processes in terms of ideles, rather than of idele-classes, preserving thus all principal idele factors which were discarded in [6]. This construction, involving the theorem of everywhere splitting algebras, the Hilbert-Speiser theorem and its idble analogy, leads to a. Though fairly explicit, these constructions of a and the Teichmfiller-

Let [Formula: see text] be a group. A set [Formula: see text] in [Formula: see text] is said to be non-commuting if [Formula: see text] for any two distinct elements [Formula: see text] and [Formula: see text] in [Formula: see text]. If [Formula: see text] for any other non-commuting set [Formula: see text] in [Formula: see text], then [Formula: see text] is said to be a maximal non-commuting set. In this paper, we obtain the cardinality of a maximal non-commuting set in [Formula: see text]-generated [Formula: see text]-groups of class two having finite commutator subgroup.

We investigate the power of counting in Group Isomorphism. We first leverage the count-free variant of the Weisfeiler–Leman Version I algorithm for groups [J. Brachter and P. Schweitzer, On the Weisfeiler–Leman dimension of finite groups, in 35th Annual ACM/IEEE Symp. Logic in Computer Science, eds. H. Hermanns, L. Zhang, N. Kobayashi and D. Miller, Saarbrucken, Germany, July 8–11, 2020 (ACM, 2020), pp. 287–300, doi:10.1145/3373718.3394786] in tandem with bounded non-determinism and limited counting to improve the parallel complexity of isomorphism testing for several families of groups. These families include: • Direct products of non-Abelian simple groups. • Coprime extensions, where the normal Hall subgroup is Abelian and the complement is an [Formula: see text]-generated solvable group with solvability class [Formula: see text]. This notably includes instances where the complement is an [Formula: see text]-generated nilpotent group. This problem was previously known to be in [Formula: see text] [Y. Qiao, J. M. N. Sarma and B. Tang, On isomorphism testing of groups with normal Hall subgroups, in Proc. 28th Symp. Theoretical Aspects of Computer Science, Dagstuhl Castle, Leibniz Center for Informatics, 2011), pp. 567–578, doi:10.4230/LIPIcs. STACS.2011.567], and the complexity was recently improved to [Formula: see text] [J. A. Grochow and M. Levet, On the parallel complexity of group isomorphism via Weisfeiler–Leman, in 24th Int. Symp. Fundamentals of Computation Theory, eds. H. Fernau and K. Jansen, Lecture Notes in Computer Science, Vol. 14292, September 18–21, 2023, Trier, Germany (Springer, 2023), pp. 234–247]. • Graphical groups of class 2 and exponent [Formula: see text] [A. H. Mekler, Stability of nilpotent groups of class 2 and prime exponent, J. Symb. Logic 46(4) (1981) 781–788] arising from the CFI and twisted CFI graphs [J.-Y. Cai, M. Fürer and N. Immerman, An optimal lower bound on the number of variables for graph identification, Combinatorica 12(4) (1992) 389–410], respectively. In particular, our work improves upon previous results of Brachter and Schweitzer [On the Weisfeiler–Leman dimension of finite groups, in 35th Annual ACM/IEEE Symp. Logic in Computer Science, eds. H. Hermanns, L. Zhang, N. Kobayashi and D. Miller, Saarbrucken, Germany, July 8–11, 2020 (ACM, 2020), pp. 287–300, doi:10.1145/3373718.3394786]. Notably, each of these families was previously known to be identified by the counting variant of the more powerful Weisfeiler–Leman Version II algorithm. We finally show that the q-ary count-free pebble game is unable to even distinguish Abelian groups. This extends the result of Grochow and Levet (ibid), who established the result in the case of [Formula: see text]. The general theme is that some counting appears necessary to place [Formula: see text] into [Formula: see text].