On the number of Abelian groups of a given order and on certain related multiplicative functions

Let a(n) denote the number of nonisomorphic Abelian groups with n elements. Several problems concerning a(n) and a class of related multiplicative functions are discussed. These include precise characterizations of local densities, distribution of values (where the Tauberian theorem of Hardy-Ramanujan is used), sharp formulas for the sums Σ_n≤xF(a(n)) for a wide class of functions F, and a comparison of values taken on the average by a(n) and some other common arithmetic functions.     

On the number of elements of given order in a finitep-group

A. Kulakoff [9] proved that forp>2 the numberN _k =N _k (G) of solutions of the equationx ^{p k} =e in a non-cyclicp-groupG is divisible byp ^k+1. This result is a generalization of the well-known theorem of G. A. Miller asserting that the numberC _k =C _k (G) of cyclic subgroups of orderp ^k >p>2 is divisible byp. In this note we show that, as a rule: (1) ifk>1, thenN _k ≡0(modp ^k+p ); (2) ifk>2, thenC _k ≡0(modp ^p ). These facts are generalizations of many results from [1–5,8,9].     

On the possible number of elements of given order in a finite group

The main motivation of this paper is to introduce a problem of some combinatorial flavor about finite groups which seems to be new in the literature. Letk>1 be a fixed positive integer and denote byf(k, G) the number of elements of orderk in the groupG. We examine the setF(k)={f(k, G)| G a finite group}/{0}. We give a complete characterization ofF(k) if 4|k ork=6 and show some modest partial results for certain other values ofk. It seems to us that the question is surprisingly difficult even in such simple cases ask=3, which we investigate in detail. Research (partially) supported by Hungarian National Foundation for Scientific Research (OTKA), Grant No. 1901. Research (partially) supported by Hungarian National Foundation for Scientific Research (OTKA), Grant No. 1903.     

Finite groups with a given number of relative centralizers

We will study all finite groups and their subgroups with at most four relative centralizers.     

On permutations of order dividing a given integer

We give a detailed analysis of the proportion of elements in the symmetric group on n points whose order divides m, for n sufficiently large and m≥n with m=O(n).      

On the maximum number of copies of H in graphs with given size and order

We study the maximum number $ex\left(n,e,H\right)$ of copies of a graph $H$ in graphs with a given number of vertices and edges. We show that for any fixed graph $H,ex\left(n,e,H\right)$ is asymptotically realized by the quasi‐clique provided that the edge density is sufficiently large. We also investigate a variant of this problem, when the host graph is bipartite.     

On the number of homogeneous models of a given power

It is shown that, For each complete theoryT, the nomberh _T(m) of homogeneous models ofT of powerm is a non-increasing function of uncountabel cardinalsm Moreover, ifh _T(ℵ₀)≦ℵ₀, then the functionh _T is also non-increasing ℵ₀ to ℵ₁. This work was supported in part by NSF contracts GP 4257 and GP5913.     

An expression for the number of subgroups of a given order of a finite p-group

We find an exact expression for the number of subgroups of any (possible) order of an arbitrary finite p-group.Translated from Matematicheskie Zametki, Vol. 12, No. 5, pp. 561–568, November, 1972.     

On the number of graphs with a given endomorphism monoid

For a given finite monoid , let be the number of graphs on n vertices with endomorphism monoid isomorphic to . For any nontrivial monoid we prove that where and are constants depending only on with .For every k there exists a monoid of size k with , on the other hand if a group of unity of has a size k>2 then .     

On finite groups whose every proper normal subgroup is a union of a given number of conjugacy classes

Let G be a finite group andA be a normal subgroup ofG. We denote by ncc(A) the number ofG-conjugacy classes ofA andA is calledn-decomposable, if ncc(A)= n. SetK _G = {ncc(A)|A ⊲ G}. LetX be a non-empty subset of positive integers. A groupG is calledX-decomposable, ifK _G =X. Ashrafi and his co-authors [1-5] have characterized theX-decomposable non-perfect finite groups forX = {1, n} andn ≤ 10. In this paper, we continue this problem and investigate the structure ofX-decomposable non-perfect finite groups, forX = {1, 2, 3}. We prove that such a group is isomorphic to Z₆, D₈, Q₈, S₄, SmallGroup(20, 3), SmallGroup(24, 3), where SmallGroup(m, n) denotes the mth group of ordern in the small group library of GAP [11].     

On the average of the number of imaginary quadratic fields with a given class number

Let \(\mathcal {F}(h)\) be the number of imaginary quadratic fields with class number h. In this note, we improve the error term in Soundararajan’s asymptotic formula for the average of \(\mathcal {F}(h)\). Our argument leads to a similar refinement of the asymptotic for the average of \(\mathcal {F}(h)\) over odd h, which was recently obtained by Holmin, Jones, Kurlberg, McLeman and Petersen.     

On a loss system with a given number of customers

A one-server loss system with Poisson arrival stream and deterministic service times is considered conditional on the number of customers who appeared up to a givenT. This condition implies that the arrival times form a sample of the uniform distribution on (0,T]. We derive several characteristics of interest, such as the blocking probability at any given timet (0,T], the probability that exactlyi of the customers in (0,T] are served and, as a generalization, the distribution of the number of served customers arriving in any subinterval of (0,T].     

On the number of subgroups of given structure in a finitep-group

Supported in part by the Ministry of Science and Technology of Israel and the Rashi Foundation.     

On the number of certain subgraphs contained in graphs with a given number of edges

All graphs considered are finite, undirected, with no loops, no multiple edges and no isolated vertices. For two graphsG, H, letN(G, H) denote the number of subgraphs ofG isomorphic toH. Define also, forl≧0,N(l, H)=maxN(G, H), where the maximum is taken over all graphsG withl edges. We determineN(l, H) precisely for alll≧0 whenH is a disjoint union of two stars, and also whenH is a disjoint union ofr≧3 stars, each of sizes ors+1, wheres≧r. We also determineN(l, H) for sufficiently largel whenH is a disjoint union ofr stars, of sizess ₁≧s ₂≧…≧s _r>r, provided (s ₁−s _r)²<s ₁+s _r−2r. We further show that ifH is a graph withk edges, then the ratioN(l, H)/l ^k tends to a finite limit asl→∞. This limit is non-zero iffH is a disjoint union of stars.