Finite Abelian P-group Research Articles

Fusion Systems over Mp(1, 1, 1) × A-groups

In this paper we study the saturated fusion systems over a direct product of the extraspecial group of order p3 of exponent p and a finite abelian p-group. The result provides some new exotic fusion systems, whose unique components are isomorphic to the exotic fusion systems over $$7_ + ^{1 + 2}$$ found by Ruiz and Viruel.

Random Partitions and Cohen–Lenstra Heuristics

We investigate combinatorial properties of a family of probability distributions on finite abelian p-groups. This family includes several well-known distributions as specializations. These specializations have been studied in the context of Cohen-Lenstra heuristics and cokernels of families of random p-adic matrices.

P-adic estimates of the number of permutation representations

Let p be a prime, and let G be a finite abelian p-group of type λ=(λ1,λ2,…) with λ1≥λ2≥⋯ and ∑λi=s. Set u=max⁡{λ1,[(s+1)/2]} and v=s−u. For each nonnegative integer n, let hn(G) be the number of homomorphisms from G to the symmetric group Sn on n letters. Except for the case where p=2 and u+δv0≤v+1, δ the Kronecker delta, or p=3 and u=v≥1, there exist p-adic analytic functions fr(X) for r=0,1,…,pu+1−1 and a polynomial η(X) with integer coefficients such that for any nonnegative integer y, hpu+1y+r(G)=p{∑j=1upj−(u−v)}yfr(y)∏j=1yη(j) and ordp(hpu+1y+r(G))={∑j=1upj−(u−v)}y+ordp(fr(y)). If p=2, λ3=0, and u=v≥1 or if p=3 and u=v≥1, then hn(G) has analogous properties. Under the assumption that λ3=0, some results for the number of permutation representations of G in the wreath product of a cyclic group of order p with Sn are also presented.

On the Structures of K2(ℤ[G]), G a Finite Abelian p-Group

Let G be a finite abelian p-group, Γ the maximal ℤ-order of ℤ[G]. We prove that the 2-primary torsion subgroups of K2(ℤ[G]) and K2(Γ) are isomorphic when p ≡ 3, 5, 7 (mod 8), and [Formula: see text] is isomorphic to [Formula: see text] when p ≡ 2, 3, 5, 7. As an application, we give the structure of K2(ℤ[G]) for G a cyclic p-group or an elementary abelian p-group.

On zero-sum subsequences in a finite abelian p-group of length not exceeding a given number

Let G be a finite abelian group. For any integer a≥1, we define the constant s≤a(G) as the least positive integer t such that any sequence S over G of length at least t has a zero-sum subsequence of length ≤a in it. In this article, we compute this constant for many classes of abelian p-groups. In particular, it proves a conjecture of Schmid and Zhuang [20].

On the Invariant Factors of Class Groups in Towers of Number Fields

AbstractFor a finite abelian p-group A of rank d = dim A/pA, let A := be its (logarithmic) mean exponent. We study the behavior of themean exponent of p-class groups in pro-p towers L/K of number fields. Via a combination of results from analytic and algebraic number theory, we construct infinite tamely ramified pro-p towers in which the mean exponent of p-class groups remains bounded. Several explicit examples are given with p = 2. Turning to group theory, we introduce an invariant attached to a finitely generated pro-p group G; when G = Gal(L/K), where L is the Hilbert p-class field tower of a number field K, measures the asymptotic behavior of the mean exponent of p-class groups inside L/K. We compare and contrast the behavior of this invariant in analytic versus non-analytic groups. We exploit the interplay of group-theoretical and number-theoretical perspectives on this invariant and explore some open questions that arise as a result, which may be of independent interest in group theory.

How Many Units Can a Commutative Ring Have?

László Fuchs posed the following problem in 1960, which remains open: determine whether a given abelian group can occur as the group of units in a commutative ring. In this note, we provide an elementary solution to a simpler, related problem: find all cardinal numbers occurring as the cardinality of the group of units in a commutative ring. As a byproduct, we obtain a solution to Fuchs's problem for the class of finite abelian p-groups when p is an odd prime.

Rationality of the zeta function of the subgroups of abelian $p$-groups

Given a finite abelian p-group F, we prove an efficient recursive formula for sigma(alpha)(F) = Sigma(H <= F) vertical bar H vertical bar(alpha) where H ranges over the subgroups of F. We infer from this formula that the p-component of the corresponding zeta-function on groups of p-rank bounded by some constant r is rational with a simple denominator. We also provide two explicit examples in rank r = 3 and r = 4, as well as, a closed formula for sigma(alpha)(F).

Pseudo-free families of finite computational elementary abelian p-groups

AbstractWe initiate the study of (weakly) pseudo-free families of computational elementary abelian

Determining of a finite Abelian p-group up to isomorphism

Determining of a finite Abelian p-group up to isomorphism

Automorphism Groups of Finite Abelian p-groups and Transvections

We define a particular type of automorphisms called transvections on a finite finite abelian p-group Hp. It is proved that the subgroup E of the automorphism group Aut(Hp) of Hp generated by those transvections is normal in it, and that Aut(Hp) can be written as the product of E and some abelian subgroup K. The center of Aut(Hp) is also determined.

Simple Functors of Admissible Linear Categories

Generalizing an idea used by Bouc, Thevenaz, Webb and others, we introduce the notion of an admissible R-linear category for a commutative unital ring R. Given an R-linear category \(\mathcal {L}\), we define an \(\mathcal {L}\)-functor to be a functor from \(\mathcal {L}\) to the category of R-modules. In the case where \(\mathcal {L}\) is admissible, we establish a bijective correspondence between the isomorphism classes of simple functors and the equivalence classes of pairs (G, V) where G is an object and V is a module of a certain quotient of the endomorphism algebra of G. Here, two pairs (F, U) and (G, V) are equivalent provided there exists an isomorphism F ← G effecting transport to U from V. We apply this to the category of finite abelian p-groups and to a class of subcategories of the biset category.

The number of automorphism of a finite abelian group of rank two

In this paper, (1) we deduce a formula for the number of automorphism of finite Abelian p-group of rank two and (2) a formula for the number of auto-morphism of finite Abelian group of rank two Zm×Zn, by using partition of number of cyclic subgroups of group .

Truncated versions of Dwork's lemma for exponentials of power series and p-divisibility of arithmetic functions

(Dieudonné and) Dwork's lemma gives a necessary and sufficient condition for an exponential of a formal power series S(z) with coefficients in Qp to have coefficients in Zp. We establish theorems on the p-adic valuation of the coefficients of the exponential of S(z), assuming weaker conditions on the coefficients of S(z) than in Dwork's lemma. As applications, we provide several results concerning lower bounds on the p-adic valuation of the number of permutation representations of finitely generated groups. In particular, we give fairly tight lower bounds in the case of an arbitrary finite Abelian p-group, thus generalising numerous results in special cases that had appeared earlier in the literature. Further applications include sufficient conditions for ultimate periodicity of subgroup numbers modulo p for free products of finite Abelian p-groups, results on p-divisibility of permutation numbers with restrictions on their cycle structure, and a curious “supercongruence” for a certain binomial sum.

The cross number of minimal zero-sum sequences in finite abelian groups

We study the maximal cross number K(G) of a minimal zero-sum sequence and the maximal cross number k(G) of a zero-sum free sequence over a finite abelian group G, defined by Krause and Zahlten. In the first part of this paper, we extend a previous result by X. He to prove that the value of k(G) conjectured by Krause and Zahlten holds for G⨁Cpa⨁Cpb when it holds for G, provided that p and the exponent of G are related in a specific sense. In the second part, we describe a new method for proving that the conjectured value of K(G) holds for abelian groups of the form Hp⨁Cqm (where Hp is any finite abelian p-group) and Cp⨁Cq⨁Cr for any distinct primes p,q,r. We also give a structural result on the minimal zero-sum sequences that achieve this value.

Explicit embeddings of finite abelian p-groups in the group $\mathcal{J}(\mathbb{F}_p )$

For every finite abelian p-group, an explicit embedding of this group in the Jennings group $\mathcal{J}(\mathbb{F}_p )$ is constructed.

Modeling the distribution of ranks, Selmer groups, and Shafarevich–Tate groups of elliptic curves

Using maximal isotropic submodules in a quadratic module over Z_p, we prove the existence of a natural discrete probability distribution on the set of isomorphism classes of short exact sequences of co-finite type Z_p-modules, and then conjecture that as E varies over elliptic curves over a fixed global field k, the distribution of 0 --> E(k) tensor Q_p/Z_p --> Sel_{p^infty} E --> Sha[p^infty] --> 0 is that one. We show that this single conjecture would explain many of the known theorems and conjectures on ranks, Selmer groups, and Shafarevich-Tate groups of elliptic curves. We also prove the existence of a discrete probability distribution of the set of isomorphism classes of finite abelian p-groups equipped with a nondegenerate alternating pairing, defined in terms of the cokernel of a random alternating matrix over Z_p, and we prove that the two probability distributions are compatible with each other and with Delaunay's predicted distribution for Sha. Finally, we prove new theorems on the fppf cohomology of elliptic curves in order to give further evidence for our conjecture.

On the explicit structure of [formula omitted] for G a finite abelian group and [formula omitted] a finite field

Let F be a finite field of characteristic p. We first determine the structure of the relative K2 group of some truncated polynomial rings over FG with G a finite abelian p-group through calculation of Dennis–Stein symbols, then we determine the explicit structure of K2(FG) with G a finite abelian group.

RELATIVE RELATION MODULES OF FINITE ELEMENTARY ABELIAN p-GROUPS

Let E be a free product of a finite number of cyclic groups, and S a normal subgroup of E such that <TEX>$$E/S{\sim_=}G$$</TEX> is finite. For a prime p, <TEX>$\hat{S}=S/S^{\prime}S^p$</TEX> may be regarded as an <TEX>$F_pG$</TEX>-module via conjugation in E. The aim of this article is to prove that <TEX>$\hat{S}$</TEX> is decomposable into two indecomposable modules for finite elementary abelian p-groups G.

Saturated fusion systems over a class of finite p-groups

Let p be an odd prime and F be a saturated fusion system over a finite p-group S with derived subgroup of prime order, excepting the case when S≅P×A where P is a minimal nonabelian p-group with P′∩℧1(P)=1, ℧1(P) is cyclic, and A is a finite abelian p-group. In this paper, we prove that S⊴F. That is, S is resistant. This generalizes a result of Stancu in the odd prime case.