FC-Groups whose central factor group can be embedded in a direct product of finite groups

We consider the question of whether an FC-group G in which the derived subgroup [G, G] is a subgroup of a direct product of finite groups must have its central factor group G/Z(G) also embeddable in a direct product of finite groups.     

On the periodic groups saturated by direct products of finite simple groups

We prove the local finiteness of a periodic group G saturated by direct products of an elementary abelian 2-group of fixed order and the simple groups L ₂(q) under condition that G contains an element of order 4.     

On the periodic groups saturated by direct products of finite simple groups. II

We continue the study that was started in [1] of periodic groups saturated with direct products of elementary abelian 2-groups and simple linear groups of dimension 2.     

Automorphisms of direct products of finite groups

This paper shows that if H and K are finite groups with no common direct factor and G = H × K, then the structure and order of Aut G can be simply expressed in terms of Aut H, Aut K and the central homomorphism groups Hom (H, Z(K)) and Hom (K, Z(H)). Received: 18 April 2005; revised: 9 June 2005     

Every finite lattice can be embedded in a finite partition lattice

Conclusions There are many questions, which arise in connection with the theorem presented. In general, we would like to know more about the class of embeddings of a given lattice in the lattices of all equivalences over finite sets. Some of these problems are studied in [4]. In this paper, an embedding is called normal, if it preserves 0 and 1. Using regraphs, our result can be easily improved as follows: THEOREM.For every lattice L, there exists a positive integer n ₀,such that for every n≥n ₀,there is a normal embedding π: L→Eq(A), where |A|=n. Embedding satisfying special properties are shown in Lemma 3.2 and Basic Lemma 6.2. We hope that our method of regraph powers will produce other interesting results. There is also a question about the effectiveness of finding an embedding of a given lattice. In particular, the proof presented here cannot be directly used to solve the following. Problem. Can the dual of Eq(4) be embedded into Eq(2¹⁰⁰⁰)? Presented by G. Gr?tzer.     

Residually finite algorithmically finite groups,their subgroups and direct products

We construct a finitely generated infinite recursively presented residually finite algorithmically finite group G, thus answering a question of Myasnikov and Osin. The group G here is “strongly infinite” and “strongly algorithmically finite,” which means that G contains an infinite Abelian normal subgroup and all finite Cartesian powers of G are algorithmically finite (i.e., for any n, there is no algorithm writing out infinitely many pairwise distinct elements of the group G ⁿ ). We also formulate several open questions concerning this topic.     

Disjoint finite partial steiner triple systems can be embedded in disjoint finite steiner triple systems

This paper shows that a pair of disjoint finite partial Steiner triple systems can be embedded in a pair of disjoint finite Steiner triple systems.     

ZL-amenability and characters for the restricted direct products of finite groups

Let G   be a restricted direct product of finite groups _{Gi}i∈I${\{G_i\}}_{i\in I}$, and let Z?¹(G)${\mathrm{Z}?}^1(G)$ denote the centre of its group algebra. We show that Z?¹(G)${\mathrm{Z}?}^1(G)$ is amenable if and only if _Gi$G_i$ is abelian for all but finitely many i  , and characterize the maximal ideals of Z?¹(G)${\mathrm{Z}?}^1(G)$ which have bounded approximate identities. We also study when an algebra character of Z?¹(G)${\mathrm{Z}?}^1(G)$ belongs to _c0$c_0$ or ?^p${?}^p$ and provide a variety of examples.     

Characterization of groups with layer-finite periodic parts

We consider groups without involutions, in which the normalizer of an arbitrary finite nontrivial subgroup, invariant with respect to a certain elementa of prime order p, has layer-finite periodic part. Under a quite weak finiteness condition it is proved that such groups also have layer-finite periodic parts.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, Nos. 7 and 8, pp. 942–946, July–August, 1991.