Semimodular lattices with isomorphic graphs

Two discrete modular lattice and have isomorphic graphs if and only if is of the form A × and is of the form A × for some lattices A and and . We prove that for discrete semimodular lattices and this latter condition holds if and only if and have isomorphic graphs and the isomorphism preserves the order on all cover-preserving sublattices of which are isomorphic to the seven-element, semimodular, nonmodular lattice (see Figure 1). This answers in the affirmative a question posed by J. Jakubik.     

Finite groups whose character degree graphs coincide with their prime graphs

In the literature, there are several graphs related to a finite group G. Two of them are the character degree graph, denoted by ΔG), and the prime graph ΓG), In this paper we classify all finite groups whose character degree graphs are disconnected and coincide with their prime graphs. As a corollary, we find all finite groups whose character degree graphs are square and coincide with their prime graphs.     

Primary Abelian groups with isomorphic endomorphism groups

A complete answer is given to the question, does isomorphism of the endomorphism groups of primary Abelian groups imply isomorphism of the groups themselves ? It is presumed that the generalized continuum hypothesis is valid.     

On groups with isomorphic holomorphs

Groups with isomorphic holomorphs are said to be holomorphically isomorphic. The following problems are treated.

1. What Abelian groups have the property that their holomorphic isomorphism implies the isomorphism of the groups themselves?
2. Does holomorphic isomorphism of two groups, one of which is Abelian, imply the commutativity of the other group?

Classes of groups are selected for which the answers to these questions are positive.     

Arithmetic groups with isomorphic finite quotients

Two finitely generated groups have the same set of finite quotients if and only if their profinite completions are isomorphic. Consider the map which sends (the isomorphism class of) an S-arithmetic group to (the isomorphism class of) its profinite completion. We show that for a wide class of S-arithmetic groups, this map is finite to one, while the fibers are of unbounded size.     

Finite groups whose conjugacy class graphs have few vertices

In this note we classify the finite groups satisfying the following property P₅: their conjugacy class lengths are set-wise relatively prime for any 5 distinct classes.Received: 6 October 2004; revised: 16 November 2004     

Decompositions of complete graphs into isomorphic cubes

It is shown that, for any positive integer d, the d-dimensional cube W_d has an α-valuation. This implies that any complete graph K_2cn+1 (where n = d2^d-1 is the number of edges of W_d and c is an arbitrary positive integer) can be decomposed into edge disjoint copies of W_d.     

Partitioning circulant graphs into isomorphic linear forests

1.IntroductionAlinearforestisaforestwhosecomponentsarepaths.Akiyama,E-coo,andHararyprovedthefollowingTheoremAll]andTheoremBIZ].TheoremA.Every3-regUlargraphGhasapartition(FI,F2)ofE(G)suchthatboth(V(G),FI)and(V(G),F2)arelinearforests.TheoremB.Every4reg...     

On isomorphic linear partitions in cubic graphs

A linear forest is a graph whose connected components are chordless paths. A linear partition of a graph G is a partition of its edge set into linear forests and la(G) is the minimum number of linear forests in a linear partition. It is well known that la(G)=2 when G is a cubic graph and Wormald [N. Wormald, Problem 13, Ars Combinatoria 23(A) (1987) 332-334] conjectured that if |V(G)|≡0 (mod 4), then it is always possible to find a linear partition in two isomorphic linear forests. Here, we give some new results concerning this conjecture.