On diagrams for abelian groups

Two operators or two subgroups of a group will be called I-conjugate whenever they are conjugate under the group of isomorphisms of this group. It is well known that all the cyclic subgroups of highest order contained in any abelian group are I-conjugate. A necessary and sufficient condition that all the cyclic subgroups of any other order contained in an abelian group G of order pmm, p being a prime number, are I-conjugate is that all the invariants of G are equal to each other. When this condition is satisfied we shall prove, in particular, that the number of the subgroups of order pa, a c m , contained in G is equal to the number of its subgroups of index pa. It is known that the ?-subgroup of G is composed of the pth power of every operator of G. The concept of +-subgroup in connection with abelian groups can readily be extended by considering the characteristic subgroup of G composed of the path power of every operator of G. This subgroup may be called the Oa-subgroup of G. It is composed of all the operators of G which are found in each one of its subgroups of index pa, and all the operators of highest order in a 4.-subgroup of G are I-conjugate. In particular, the 0&-subgroup generated by operators of order p is the fundamental characteristic subgroup of G. t In fact, when p > 2 the 0a-subgroup which is generated by operators of order pr is contained in every characteristic subgroup of G which involves operators of order pr. The quotient group of G corresponding to a 4a-subgroup may be called a 0,fi-quotient group. This quotient group is simply isomorphic with the subgroup of G generated bv all its operators whose orders divide p> Hence the following: THEOREM. The number of the subgroups of index pa contained in any abelian

We prove that a group G is Abelian whenever (1) it is nilpotent and the lattice of normal subgroups of G is isomorphic to the subgroup lattice of an Abelian group or (2) there exists a non-torsion Abelian group B such that the normal subgroup lattice of B × G is isomorphic to the subgroup lattice of an Abelian group. Using (2), it is proved that an Abelian group A can be determined in the class of all groups by the lattice of all normal subgroups of some groups, e.g. if A is an Abelian group and G is a group such that \mathbb Z × A and \mathbb Z × G have isomorphic normal subgroup lattices then A and G are isomorphic groups.

In 1908 Professor G. A. Miller showed that if an abelian group H which involves operators whose orders exceed 2 is extended by means of an operator of order 2 which transforms each operator of H into its inverse, then the group of isomorphisms of this extended group is the holomorph of H.t The present paper discusses an elaboration of the idea embodied in Professor Miller's thoerem, the successive developments taking an abelian group H each of more general type so that in toto it is proved that if G is formed by extending H which has operators of order > 2 by a certain operator from its group of isomorphisms which transforms every one of its operators into the same power of itself and which is commutative with no operator of odd order in it, then the group of isomorphisms of G is the holomorph of H, and is a complete group if H is of odd order. If H contains no operators of even order, the certain operator is any operator from the group of isomorphisms of H effecting the stated automorphism; if H contains no operator of odd order, the certain operator must transform every operator of H into its inverse; if H contains operators of both even and odd orders (a) the order of the automorphism of the operators of odd order effected by the extending operator is to be divisible by 2n where H contains an operator of order 2n but none of order 2n+1, or else (b) the extending operator transforms into its inverse every one of H's operators whose order is a power of 2. Obviously it is not necessary that the extending operator be from its group of isomorphisms, but that it have certain properties possessed by this operator; thus, besides effecting the automorphism required, its first power commutative with all the operators of H must be the identity (from which follows that its first power appearing in H is the identity). The general method used in establishing each of the successive theorems is the same; two important steps in each proof are showing that H is character-

Introduction. In [7] and [8] we have determined the isomorphisms of the endomorphism rings of two special classes of torsion-free modules. In [7] the modules were free, over principal left ideal domains, and the isomorpnism was always induced by a semi-linear transformation of the modules. In [8] the ring of scalars was a complete discrete valuation ring, but no further restriction was placed on the modules. Except for the special case when both modules are divisible, the same result holds as in [7]. In each case a key point in the argument is that a certain indecomposable direct summand can be shown to be cyclic. Even for ordinary torsion-free abelian groups, the abundance of the indecomposable ones makes it impossible to prove that isomorphism of the endomorphism rings implies isomorphism of the underlying groups. We show first that for any cardinal number c, there exist homogeneous completely decomposable groups G and H of rank c which are not isomorphic, but have isomorphic rings of endomorphisms. This makes it clear that strong restrictions must be placed on the class of groups considered if it is hoped that the ring isomorphism is to be induced by a group isomorphism. In particular, homogeneity coupled with separability or complete decomposability is not sufficient. We are, however, able to generalize the results of [7] to a class of what we call locally free modules A over principal left ideal domains F. A locally free abelian group (F is the ring of integers) is just a homogeneous separable group of null type [6, p. 208]. This latter class of groups includes the complete direct sum of infinite cyclic groups, as well as all its subgroups [4, p. 176]. For example, the set of all bounded sequences of integers is such a group. We also determine (Theorem B) which semi-linear transformations induce the same ring isomorphisms. The results are similar to the vector space case considered in [2], and apply to the endomorphism rings of locally free modules as well as the torsion-free modules of [8].

In the paper the concept of the similarity of homogeneously decomposable groups is introduced. One investigates the cases in which almost isomorphic groups are similar or isomorphic.

New constructions of strongly regular Cayley graphs on abelian non p-groups

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].

Let X be a separated scheme of finite type over an algebraically closed field k and let m be a natural number. By an explicit geometric construction using torsors we construct a pairing between the first mod m Suslin homology and the first mod m tame étale cohomology of X . We show that the induced homomorphism from the mod m Suslin homology to the abelianized tame fundamental group of X mod m is surjective. It is an isomorphism of finite abelian groups if (m, char(k)) = 1 , and for general m if resolution of singularities holds over k .

In this paper the stability of the quadratic equation is considered on arbitrary groups. Since the quadratic equation is stable on Abelian groups, this paper examines the stability of the quadratic equation on noncommutative groups. It is shown that the quadratic equation is stable on $n$-Abelian groups when $n$ is a positive integer. The stability of the quadratic equation is also established on the noncommutative group $T(2, K)$, where $K$ is an arbitrary commutative field. It is proved that every group can be embedded into a group in which the quadratic equation is stable.

Given an abelian group G of order n, and a finite non-empty subset A of integers, the Davenport constant of G with weightA, denoted by DA(G), is defined to be the least positive integer t such that, for every sequence (x1,..., xt) with xi ∈ G, there exists a non-empty subsequence \((x_{j_1},\ldots, x_{j_l})\) and ai ∈ A such that \(\sum_{i=1}^{l}a_ix_{j_i} = 0\). Similarly, for an abelian group G of order n, EA(G) is defined to be the least positive integer t such that every sequence over G of length t contains a subsequence \((x_{j_1} ,\ldots, x_{j_n})\) such that \(\sum_{i=1}^{n}a_ix_{j_i} = 0\), for some ai ∈ A. When G is of order n, one considers A to be a non-empty subset of {1,..., n − 1 }. If G is the cyclic group \({\Bbb Z}/n{\Bbb Z}\), we denote EA(G) and DA(G) by EA(n) and DA(n) respectively.

The aim of this paper is to determine the structure and to establish the isomorphic invariant of the finitely generated nilpotent group G of infinite cyclic commutator subgroup. Using the structure and invariant of the group which is the central extension of a cyclic group by a free abelian group of finite rank of infinite cyclic center, we provide a decomposition of G as the product of a generalized extraspecial ℤ-group and its center. By using techniques of lifting isomorphisms of abelian groups and equivalent normal form of the generalized extraspecial ℤ-groups, we finally obtain the structure and invariants of the group G.

In the previous chapter, we dealt with solvable groups and the methods came from abelian groups. In this chapter, we deal with \(S{L_2}({\mathcal{O}_S})\) where \(\mathcal{O}\) is the ring of integers in an algebraic number field and S is any set of places containing all the archimedean ones. The methods here will be very different from those of the previous chapter.

Minimal sumsets in finite solvable groups

All groups herein are assumed to be abelian. It was not until the 1940's that it was known that a subgroup of an infinite direct sum of finite cyclic groups is again a direct sum of cyclics. This result rests on a general criterion due to Kulikov [7] for a primary abelian group to be a direct sum of cyclic groups. If G is p-primary, Kulikov's criterion presupposes that G has no elements (other than zero) having infinite p-height. For such a group G, the criterion is simply that G be the union of an ascending sequence of subgroups Hn where the heights of the elements of Hn computed in G are bounded by some positive integer λ(n). The theory of abelian groups has now developed to the point that totally projective groups currently play much the same role, at least in the theory of torsion groups, that direct sums of cyclic groups and countable groups played in combination prior to the discovery of totally projective groups and their structure beginning with a paper by R. Nunke [11] in 1967.

It is shown that a subgroup H H of an abelian group G G is an intersection of pure subgroups of G G if and only if, for all primes p p and positive integers n n , p n g ∈ H {p^n}g \in H and p n − 1 g ∉ H {p^{n - 1}}g \notin H imply that there exists z ∈ G z \in G such that p n z = 0 {p^n}z = 0 and p n − 1 z ∉ H {p^{n - 1}}z \notin H . This solves a problem posed by L L . Fuchs in [2] and [3].