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]).     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> approximation capability of RBF neural networks

L ^p approximation capability of radial basis function (RBF) neural networks is investigated. If g: R ₊¹ → R ¹ and ∈ L _loc ^p (R ⁿ ) with 1 ≤ p < ∞, then the RBF neural networks with g as the activation function can approximate any given function in L ^p (K) with any accuracy for any compact set K in R ⁿ , if and only if g(x) is not an even polynomial. Partly supported by the National Natural Science Foundation of China (10471017)     

Weighted <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-conjecture for locally compact groups

Let G be a locally compact group, ω be a weight function on G and 1 < p < ∞. Here, we give a sufficient condition for that the weighted L ^p -space L ^p (G, ω) is a Banach algebra. Also, we get some necessary conditions on G and the weight function ω for L ^p (G, ω) to be a Banach algebra. As a consequence, we show that if G is abelian and L ^p (G, ω) is a Banach algebra, then G is σ-compact.     

<Emphasis Type="Italic">p</Emphasis>-groups having few almost-rational irreducible characters

We prove that a 2-group has exactly five rational irreducible characters if and only if it is dihedral, semidihedral or generalized quaternion. For an arbitrary prime p, we say that an irreducible character χ of a p-group G is “almost rational” if ℚ(χ) is contained in the cyclotomic field ℚ _p , and we write ar(G) to denote the number of almost-rational irreducible characters of G. For noncyclic p-groups, the two smallest possible values for ar(G) are p ² and p ² + p − 1, and we study p-groups G for which ar(G) is one of these two numbers. If ar(G) = p ² + p − 1, we say that G is “exceptional”. We show that for exceptional groups, |G: G′| = p ², and so the assertion about 2-groups with which we began follows from this. We show also that for each prime p, there are exceptional p-groups of arbitrarily large order, and for p ≥ 5, there is a pro-p-group with the property that all of its finite homomorphic images of order at least p ³ are exceptional.     

On the <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-conjecture for locally compact groups

Let G be a locally compact group. For 1 < p < ∞, it is well-known that f * g exists and belongs to L^p(G) for all f, g L^p (G) if and only if G is compact. Here, for 2 < p < ∞, we show that f * g exists for all f, g L^p(G) if and only if G is compact. We also show that this result does not remain true for 1 < p ≤ 2. Received: 23 April 2006     

On the nonabelian tensor square and capability of groups of order <Emphasis Type="Italic">p</Emphasis><Superscript>2</Superscript><Emphasis Type="Italic">q</Emphasis>

A group G is said to be capable if it is isomorphic to the central factor group H/Z(H) for some group H. Let G be a nonabelian group of order p ² q for distinct primes p and q. In this paper, we compute the nonabelian tensor square of the group G. It is also shown that G is capable if and only if either Z(G) = 1 or p < q and G^ab=\mathbbZ_p×\mathbbZ_p{G^{\rm ab}=\mathbb{Z}_{p}\times\mathbb{Z}_{p}} .     

Upper triangular operators and <Emphasis Type="Italic">p</Emphasis>-adic <Emphasis Type="Italic">L</Emphasis>-functions

For a newform f for Γ₀(N) of even weight k supersingular at a prime p ≥ 5, by using infinite dimensional p-adic analysis, we prove that the p-adic L-function L _p (f,α; χ) has finite order of vanishing at any character of the form [(c)\tilde] _s ( x ) = x^s\tilde \chi _s \left( x \right) = x^s. In particular, under the natural embedding of ℤ _p in the group of ℂ* _p -valued continuous characters of ℤ* _p , the order of vanishing at any point is finite.     

On <Emphasis Type="Italic">CEP</Emphasis>-subgroups of <Emphasis Type="Italic">n</Emphasis>-periodic products

There is a well-known fact, that any group G ₁ is a CEP-subgroup both for the direct product G ₁ × G ₂ and the free productG ₁ * G ₂ of G ₁ with any group G ₂. The paper gives a necessary and sufficient condition providing that a multiplier G _i of a n-periodic product Π _i∈I ⁿ G _i of any family of groups {G _i } _i∈I is a CEP-subgroup. Particularly, the found criterionmeans that any group G ₁ of odd period n ≥ 665 is a CEP-subgroup of the n-periodic product Π _i∈I ⁿ G _i for any group G ₂.     

Quasirecognition by prime graph of the simple group <Superscript>2</Superscript><Emphasis Type="Italic">F</Emphasis><Subscript>4</Subscript>(<Emphasis Type="Italic">q</Emphasis>)

As the main result, we show that if G is a finite group such that Γ(G) = Γ(² F ₄(q)), where q = 2^2m+1 for some m ≧ 1, then G has a unique nonabelian composition factor isomorphic to ² F ₄(q). We also show that if G is a finite group satisfying |G| =|² F ₄(q)| and Γ(G) = Γ(² F ₄(q)), then G ≅ ² F ₄(q). As a consequence of our result we give a new proof for a conjecture of W. Shi and J. Bi for ² F ₄(q). The third author was supported in part by a grant from IPM (No. 87200022).     

On the parity of the class number of an imaginary abelian field of conductor 2<Superscript><Emphasis Type="Italic">a</Emphasis></Superscript><Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">b</Emphasis></Superscript>

Let p be an odd prime number, and p^n₀{p^{n_0}} the highest power of p dividing 2 ^p−1 − 1. Let K_n=Q(z_pⁿ⁺¹){K_n={\bf Q}(\zeta_{p^{n+1}})} and L_n,j=K_n⁺(z_2^j+2){L_{n,j}=K_n^+(\zeta_{2^{j+2}})} for j ≥ 0. Let h_n^*{h_n^*} be the relative class number of K _n , and h _n,j the class number of L _n,j , respectively. Let n be an integer with n ≥ n ₀. We prove that if the ratio h_n^*/h_n-1^*{h_n^*/h_{n-1}^*} is odd, then h _n,j /h _n−1,j is odd for any j ≥ 0.     

On <Emphasis Type="Italic">p</Emphasis>-nilpotency and minimal subgroups of finite groups

We call a subgroup H of a finite group G c-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K ⩽ core(H). In this paper it is proved that a finite group G is p-nilpotent if G is S ₄-free and every minimal subgroup of P ∩ G ^N is c-supplemented in N _G (P), and when p = 2 P is quaternion-free, where p is the smallest prime number dividing the order of G, P a Sylow p-subgroup of G. As some applications of this result, some known results are generalized.     

<Emphasis Type="Italic">c</Emphasis>*-Supplemented subgroups and <Emphasis Type="Italic">p</Emphasis>-nilpotency of finite groups

A subgroup H of a finite group G is said to be c*-supplemented in G if there exists a subgroup K such that G = HK and H ⋂ K is permutable in G. It is proved that a finite group G that is S ₄-free is p-nilpotent if N _G (P) is p-nilpotent and, for all x ∈ G\N _G (P), every minimal subgroup of is c*-supplemented in P and (if p = 2) one of the following conditions is satisfied: (a) every cyclic subgroup of of order 4 is c*-supplemented in P, (b) , (c) P is quaternion-free, where P a Sylow p-subgroup of G and is the p-nilpotent residual of G. This extends and improves some known results. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 8, pp. 1011–1019, August, 2007.     

2-Recognizability by prime graph of <Emphasis Type="Italic">PSL</Emphasis>(2, <Emphasis Type="Italic">p</Emphasis><Superscript>2</Superscript>)

Let G be a finite group and let Γ(G) be the prime graph of G. Assume p prime. We determine the finite groups G such that Γ(G) = Γ(PSL(2, p ²)) and prove that if p ≠ 2, 3, 7 is a prime then k(Γ(PSL(2, p ²))) = 2. We infer that if G is a finite group satisfying |G| = |PSL(2, p ²)| and Γ(G) = Γ(PSL(2, p ²)) then G ? PSL(2, p ²). This enables us to give new proofs for some theorems; e.g., a conjecture of W. Shi and J. Bi. Some applications are also considered of this result to the problem of recognition of finite groups by element orders.     

On the abelianizations of congruence subgroups of Aut(<Emphasis Type="Italic">F</Emphasis><Subscript>2</Subscript>)

Let F _n be the free group of rank n, and let Aut⁺(F _n ) be its special automorphism group. For an epimorphism π : F _n → G of the free group F _n onto a finite group G we call the standard congruence subgroup of Aut⁺(F _n ) associated to G and π. In the case n = 2 we fully describe the abelianization of Γ⁺(G, π) for finite abelian groups G. Moreover, we show that if G is a finite non-perfect group, then Γ⁺(G, π) ≤ Aut⁺(F ₂) has infinite abelianization.     

Torsion completeness of <Emphasis Type="Italic">p</Emphasis>-primary components in modular group rings of <Emphasis Type="Italic">p</Emphasis>-reduced Abelian groups

Let G be a p-reduced Abelian group and R a commutative unital ring of prime characteristic p such that for each natural number i the subring $ R^{p^i } $ R^{p^i } has nilpotent elements. It is shown that if S(RG) is the normalized Sylow p-group in the group ring RG, then S(RG) is torsion-complete if and only if G is a bounded p-group. This strengthens our former results on this subject.     

Characterization of finite simple group <Emphasis Type="Italic">D</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>(2)

Let G be a finite group, and let π _e (G) be the spectrum of G, that is, the set of all element orders of G. In 1987, Shi Wujie put forward the following conjecture. If G is a finite group and M is a non-abelian simple group, then G ≅ M if and only if |G| = |M| and π _e (G) = π _e (M). In this short paper, we prove that if G is a finite group, then G ≅ M if and only if |G| = |M| and π _e (G) = π _e (M), where M = D _n (2) and n is even.     

Isomorphic digraphs from powers modulo <Emphasis Type="Italic">p</Emphasis>

Let p be a prime. We assign to each positive number k a digraph G _p ^k whose set of vertices is {1, 2, …, p − 1} and there exists a directed edge from a vertex a to a vertex b if a ^k ≡ b (mod p). In this paper we obtain a necessary and sufficient condition for G_p^k₁ @ G_p^k₂G_p^{{k_1}} \simeq G_p^{{k_2}}.     

Fixed-point-free elements in <Emphasis Type="Italic">p</Emphasis>-groups

In this paper we prove that there exists no function F(m, p) (where the first argument is an integer and the second a prime) such that, if G is a finite permutation p-group with m orbits, each of size at least p ^F(m,p), then G contains a fixed-point-free element. In particular, this gives an answer to a conjecture of Peter Cameron; see [4], [6].     

A note on <Emphasis Type="Italic">p</Emphasis>-groups of order ≤ <Emphasis Type="Italic">p</Emphasis><Superscript>4</Superscript>

In [1], we defined c(G), q(G) and p(G). In this paper we will show that if G is a p-group, where p is an odd prime and |G| ≤ p ⁴, then c(G) = q(G) = p(G). However, the question of whether or not there is a p-group G with strict inequality c(G) = q(G) < p(G) is still open.