Nonnormal and minimal nonabelian subgroups of a finite group

Let G be a finite p-group. If p = 2, then a nonabelian group G = Ω₁(G) is generated by dihedral subgroups of order 8. If p > 2 and a nonabelian group G = Ω₁(G) has no subgroup isomorphic to S_p²{\Sigma _{{p^2}}}, a Sylow p-subgroup of the symmetric group of degree p ², then it is generated by nonabelian subgroups of order p ³ and exponent p. If p > 2 and the irregular p-group G has < p nonabelian subgroups of order p ^p and exponent p, then G is of maximal class and order p ^p+1. We also study in some detail the p-groups, containing exactly p nonabelian subgroups of order p ^p and exponent p. In conclusion, we prove three new counting theorems on the number of subgroups of maximal class of certain type in a p-group. In particular, we prove that if p > 2, and G is a p-group of order > p ^p+1, then the number of subgroups ≅ ΣS_p²{\Sigma _{{p^2}}} in G is a multiple of p.     

可除阿贝尔p-群的自同构群

主要探讨了秩大于或者等于p-1的可除阿贝尔p-群的p-自同构群,并且得到这些p-自同构如何作用在该可除阿贝尔p-群上.这些结论有助于进一步理解 ?ernikov p-群的结构.     

Random and Deterministic Triangle Generation of Three-Dimensional Classical Groups II

Let p ₁, p ₂, p ₃ be primes. This is the second article in a series of three on the (p ₁, p ₂, p ₃)-generation of the finite projective special unitary and linear groups PSU₃(p ⁿ ), PSL₃(p ⁿ ), where we say a noncyclic group is (p ₁, p ₂, p ₃)-generated if it is a homomorphic image of the triangle group T _{p 1, p 2, p 3} . This paper is concerned with the case where p ₁ = 2 and p ₂ = p ₃. We determine for any prime p ₂ the prime powers p ⁿ such that PSU₃(p ⁿ ) (respectively, PSL₃(p ⁿ )) is a quotient of T = T _{2, p 2, p 2} . We also derive the limit of the probability that a randomly chosen homomorphism in Hom(T, PSU₃(p ⁿ )) (respectively, Hom(T, PSL₃(p ⁿ ))) is surjective as p ⁿ tends to infinity.     

Bounding an Index by the Largest Character Degree of a p-Solvable Group

In this article, we show that if p is a prime and G is a p-solvable group, then |G: O _p (G)| _p  ≤ (b(G) ^p /p)^1/(p?1), where b(G) is the largest character degree of G. If p is an odd prime that is not a Mersenne prime or if the nilpotence class of a Sylow p-subgroup of G is at most p, then |G: O _p (G)| _p  ≤ b(G).     

Relative ( pn , p , pn , n )-difference sets with GCD ( p , n )=1

Let p be an odd prime. We first get some non-existence and structural results on (pn,p,pn,n) relative difference sets with gcd(p,n)=1 through a group ring approach. We then give a construction of (p(p+1),p,p(p+1),p+1) relative difference sets with p a Mersenne prime. Supported by National Natural Science Foundation of China (10331030).     

Character Sums in Short Intervals and the Multiplication Table Modulo a Large Prime

We obtain nontrivial estimates of character sums over short intervals for almost all moduli. These bounds and the method of Karatsuba for solving multiplicative ternary problems are used to prove that for π(X)(1 + o(1)) primes p,p ≤ X, there are p(1 + o(1)) residue classes modulo p of the form xy (mod p), where 1 ≤ x, y ≤ p^?(log p)^1,087. We also prove that for any prime p there are p(1 + o(1)) residue classes modulo p of the form xy^* (mod p), where 1 ≤ x, y ≤ p^?(log p)^1+o(1) and y^* is defined by yy^* ≡ 1 (mod p).     

Modes and moments of unimodal distributions

Summary For a unimodal distribution relations of its modea with its absolute momentβ _p and central absolute momentγ _p of orderp are considered. The best constantA _p andB _p are given for the inequalities |a|≦A _p β _p ^1/p (p>0) and |a−m|≦B _p γ _p ^1/p (p≧1) wherem is the mean. the results follow from discussion of more general moments.     

Character Sums in Short Intervals and the Multiplication Table Modulo a Large Prime

We obtain nontrivial estimates of character sums over short intervals for almost all moduli. These bounds and the method of Karatsuba for solving multiplicative ternary problems are used to prove that for π(X)(1 + o(1)) primes p,p ≤ X, there are p(1 + o(1)) residue classes modulo p of the form xy (mod p), where 1 ≤ x, y ≤ p^?(log p)^1,087. We also prove that for any prime p there are p(1 + o(1)) residue classes modulo p of the form xy^* (mod p), where 1 ≤ x, y ≤ p^?(log p)^1+o(1) and y^* is defined by yy^* ≡ 1 (mod p).     

Finite <Emphasis Type="Italic">p</Emphasis>-central groups of height <Emphasis Type="Italic">k</Emphasis>

A finite group G is called p ⁱ -central of height k if every element of order p ⁱ of G is contained in the k ^th -term ζ _k (G) of the ascending central series of G. If p is odd, such a group has to be p-nilpotent (Thm. A). Finite p-central p-groups of height p − 2 can be seen as the dual analogue of finite potent p-groups, i.e., for such a finite p-group P the group P/Ω₁(P) is also p-central of height p − 2 (Thm. B). In such a group P, the index of P ^p is less than or equal to the order of the subgroup Ω₁(P) (Thm. C). If the Sylow p-subgroup P of a finite group G is p-central of height p − 1, p odd, and N _G (P) is p-nilpotent, then G is also p-nilpotent (Thm. D). Moreover, if G is a p-soluble finite group, p odd, and P ∈ Syl _p (G) is p-central of height p − 2, then N _G (P) controls p-fusion in G (Thm. E). It is well-known that the last two properties hold for Swan groups (see [11]).     

p-Banach Spaces and p-Totally Convex Spaces

We introduce the categories Vec _p of p-normed vector spaces, Ban _p of _p -Banach spaces, AC _p of _p -absolutely and TC _p of _p -totally convex spaces (0 < p 1). It will be shown that TC _p (AC _p ) is the Eilenberg–Moore category of Ban _p (Vec _p ). Then congruence relations on TC _p (AC _p )-spaces are studied. There are many differences between TC _p (AC _p )-spaces and totally (absolutely) convex spaces (i.e. p = 1) (Pumplün and Röhrl, 1984, 1985), which will become apparent in Section 4.     

A survey of orthogonal arrays of strength two

ＡＳＵＲＶＥＹＯＦＯＲＴＨＯＧＯＮＡＬＡＲＲＡＹＳＯＦＳＴＲＥＮＧＴＨＴＷＯＬＩＵＺＨＡＮＧＷＥＮ（刘璋温）（ＩｎｓｔｉｔｕｔｅｏｆＡｐｐｌｉｅｄＭａｔｈｅｍａｔｉｃｓ．ｔｈｅＣｈｉｎｅｓｅＡｃａｄｅｍｙｏｆＳｃｉｅｔｉｃｅｓ．Ｂｅｉｊｉｎｇ１０００...     

Prime Chains and Pratt Trees

Prime chains are sequences $p_{1}, \ldots , p_{k}Prime chains are sequences p₁, ?, p_kp_{1}, \ldots , p_{k} of primes for which p_j+1 o 1{p_{j+1} \equiv 1} (mod p _j ) for each j. We introduce three new methods for counting long prime chains. The first is used to show that N(x; p) = O_e(x^1+e){N(x; p) = O_{\varepsilon}(x^{1+\varepsilon})}, where N(x; p) is the number of chains with p ₁ = p and p_k ￡ p_x{p_k \leq p_x}. The second method is used to show that the number of prime chains ending at p is ≍ log p for most p. The third method produces the first nontrivial upper bounds on H(p), the length of the longest chain with p _k = p, valid for almost all p. As a consequence, we also settle a conjecture of Erdős, Granville, Pomerance and Spiro from 1990. A probabilistic model of H(p), based on the theory of branching random walks, is introduced and analyzed. The model suggests that for most p ￡ x{p \leq x}, H(p) stays very close to e log log x.     

The eigen functions of the complex projective space

In the complexn-dimensional projective spaceCP ⁿ , let λ _p (=4p(p+n)) be the eigen value of the Laplace-Beltrami operator andH _p be the space of all eigen functions of eigen value λ _p . The reproducing kernelh _p (z, w) ofH _p is constructed explicitly in this paper, and a system of complete orthogohal functions ofH _p is constructed fromh _p (z,w)(p=1,2, …). Partially supported by NSF of China     

3-Sasakian Geometry, Nilpotent Orbits, and Exceptional Quotients

Using 3-Sasakian reduction techniques we obtain infinite familiesof new 3-Sasakian manifolds M (p ₁,p ₂, p ₃) andM (p ₁,p ₂, p ₃, p ₄) in dimension 11 and 15 respectively. The metric cone on (p ₁,p ₂, p ₃) is a generalization ofthe Kronheimer hyperkähler metric on the regular maximalnilpotent orbit of sl (3, C)whereas the cone on M (p ₁,p ₂, p ₃, p ₄)generalizes the hyperkähler metric onthe 16-dimensional orbit of so(6, C).These are the first examples of 3-Sasakian metrics which are neither homogeneous nor toric. In addition we consider some further U(1)-reductions of M(p ₁,p ₂, p ₃).These yield examples of nontoric 3-Sasakian orbifold metrics in dimensions 7. As a result we obtain explicit families O() of compact self-dual positivescalar curvature Einstein metrics with orbifoldsingularities and with only one Killing vector field.     

Cp

The spacec _p is the class of operators on a Hilbert space for which thec _p norm |T| _p =[trace(T*T) ^p/2]^1/p is finite. We prove many of the known results concerningc _p in an elementary fashion, together with the result (new for 1<p<2) thatc _p is as uniformly convex a Banach space asl _p. In spite of the remarkable parallel of norm inequalities in the spacesc _p andl _p, we show thatp ≠ 2, noc _p built on an infinite dimensional Hilbert space is equivalent to any subspace of anyl _p orL _p space. The author was supported by National Science Foundation Grant GP-5707.     

On the corestriction ofp n -symbol

LetF be a field of characteristicp. Teichmüller proved that anyp-algebra overF of indexp ⁿ and exponentp ^e is similar to a tensor product with at mostp ⁿ !(p ⁿ !−1) factors of cyclicp-algebras overF of degreep ^e . In this note we improve Teichmüller bound for two particular types ofp-algebras. LetL be a finite separable extension ofF. IfA is a cyclicp-algebra overL of degreep ^e we show that Cor _L/F A, the corestriction ofA, is similar to a tensor product with at most [L :F] factors of cyclicp-algebras overF of degreep ^e . Moreover we prove that [L :F] is the best possible bound. From this we deduce that ifA is a cyclicp-algebra overF of degreep ⁿ and exponentp ^e thenA is similar to a tensor product with at mostp ^n−e factors of cyclicp-algebras overF of degreep ^e .     

A Series of Regular Hadamard Matrices

Let p and 2p−1 be prime powers and p ≡ 3 (mod 4). Then there exists a symmetric design with parameters (4p², 2p² − p, p² − p). Thus there exists a regular Hadamard matrix of order 4p².     

An extremal function for digraph subcontraction

We determine, to within a constant factor, the maximum size of a digraph which has no subcontraction to the complete digraph DK_p of order p. Let d(p) be defined for positive integers p by d(p) = inf{c; e(D) ≥ c |D| implies D (FANCY MORE THAN) DK_p}, where D denotes a digraph, and (FANCY MORE THAN) denotes contraction. We show that 0.53p√log₂p < d(p) ≤ 2502p√log₂p holds if p is sufficiently large. Hence the function d(p) differs by only a constant factor from the corresponding function for undirected graphs. © 1996 John Wiley & Sons, Inc.     

Embedding of C q and R q into noncommutative L p -spaces, 1≤p<q≤2

We prove that a quotient of a subspace of C_p⊕_pR_p (1≤p<2) embeds completely isomorphically into a noncommutative L_p -space, where C_p and R_p are respectively the p-column and p-row Hilbertian operator spaces. We also represent C_q and R_q (p<q≤2) as quotients of subspaces of C_p⊕_pR_p. Consequently, C_q and R_q embed completely isomorphically into a noncommutative L_p (M). We further show that the underlying von Neumann algebra M cannot be semifinite.     

On the p—Local Rank of Finite Groups

In this paper,we shall mainly study the p-solvable finite group in terms of p-local rank,and a group theoretic characterization will be given of finite p-solvabel groups with p-local rank two.Theorem A Let G be a finite p-solvable group with p-local rank plr(G)=2 and Op(G)=1.If P is a Sylow p-subgrounp of G,then P has a normal subgroup Q such that P/Q is cyclic or a generalized quaternion 2-group and the p-rank of Q is at most two.Theorem B Let G be a finite p-solvable group with Op(G)=1.Then the p-length lp(G)≤plr(G);if in addition plr(G)=lp (G) and p≥5 is odd,then plr(G)=0 or 1.