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1.
For a finite p-group G and a positive integer k let I
k
(G) denote the intersection of all subgroups of G of order p
k
. This paper classifies the finite p-groups G with Ik(G) @ Cpk-1{{I}_k(G)\cong C_{p^{k-1}}} for primes p > 2. We also show that for any k, α ≥ 0 with 2(α + 1) ≤ k ≤ n−α the groups G of order p
n
with Ik(G) @ Cpk-a{{I}_k(G)\cong C_{p^{k-\alpha}}} are exactly the groups of exponent p
n-α
. 相似文献
2.
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. 相似文献
3.
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. 相似文献
4.
Gustavo A. Fernández-Alcober Jon González-Sánchez Andrei Jaikin-Zapirain 《Israel Journal of Mathematics》2008,166(1):393-412
Let G be a pro-p group and let k ≥ 1. If γ
k(p−1) (G) ≤ γ
r
for some r and s such that k(p − 1) < r + s(p − 1), we prove that the exponent of Ωi(G) is at most p
i+k−1 for all i.
Supported by the Spanish Ministry of Science and Education, grant MTM2004-04665, partly with FEDER funds.
The first author is also supported by the University of the Basque Country, grant UPV05/99.
The second author is also supported by the Basque Government. 相似文献
5.
In this paper it is proved that ifp is a prime dividing the order of a groupG with (|G|,p − 1) = 1 andP a Sylowp-subgroup ofG, thenG isp-nilpotent if every subgroup ofP ∩G
N
of orderp is permutable inN
G
(P) and whenp = 2 either every cyclic subgroup ofP ∩G
N
of order 4 is permutable inN
G
(P) orP is quaternion-free. Some applications of this result are given.
The research of the first author is supported by a grant of Shanxi University and a research grant of Shanxi Province, PR
China.
The research of the second author is partially supported by a UGC(HK) grant #2160126 (1999/2000). 相似文献
6.
Let M be a compact manifold of dimension n, P=P(h) a semiclassical pseudodifferential operator on M, and u=u(h) an L
2 normalized family of functions such that P(h)u(h) is O(h) in L
2(M) as h↓0. Let H⊂M be a compact submanifold of M. In a previous article, the second-named author proved estimates on the L
p
norms, p≥2, of u restricted to H, under the assumption that the u are semiclassically localized and under some natural structural assumptions about the principal symbol of P. These estimates are of the form Ch
−δ(n,k,p) where k=dim H (except for a logarithmic divergence in the case k=n−2, p=2). When H is a hypersurface, i.e., k=n−1, we have δ(n,n−1, 2)=1/4, which is sharp when M is the round n-sphere and H is an equator. 相似文献
7.
A permutation groupG of finite degreed is called a sharp permutation group of type {k},k a non-negative integer, if every non-identity element ofG hask fixed points and |G|=d−k. We characterize sharp non-abelianp-groups of type {k} for allk. 相似文献
8.
All groups considered in this paper will be finite. Our main result here is the following theorem. Let G be a solvable group in which the Sylow p-subgroups are either bicyclic or of order p
3 for any p ∈ π(G). Then the derived length of G is at most 6. In particular, if G is an A4-free group, then the following statements are true: (1) G is a dispersive group; (2) if no prime q ∈ π(G) divides p
2 + p + 1 for any prime p ∈ π(G), then G is Ore dispersive; (3) the derived length of G is at most 4. 相似文献
9.
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 ) = xs\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. 相似文献
10.
11.
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 Gpk1 @ Gpk2G_p^{{k_1}} \simeq G_p^{{k_2}}. 相似文献
12.
As the main result, we show that if G is a finite group such that Γ(G) = Γ(2
F
4(q)), where q = 22m+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). 相似文献
13.
The Erdős-Sós conjecture says that a graph G on n vertices and number of edges e(G) > n(k− 1)/2 contains all trees of size k. In this paper we prove a sufficient condition for a graph to contain every tree of size k formulated in terms of the minimum edge degree ζ(G) of a graph G defined as ζ(G) = min{d(u) + d(v) − 2: uv ∈ E(G)}. More precisely, we show that a connected graph G with maximum degree Δ(G) ≥ k and minimum edge degree ζ(G) ≥ 2k − 4 contains every tree of k edges if d
G
(x) + d
G
(y) ≥ 2k − 4 for all pairs x, y of nonadjacent neighbors of a vertex u of d
G
(u) ≥ k. 相似文献
14.
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. 相似文献
15.
Štefan Gyürki 《Mathematica Slovaca》2009,59(2):193-200
Let k be an integer. A 2-edge connected graph G is said to be goal-minimally k-elongated (k-GME) if for every edge uv ∈ E(G) the inequality d
G−uv
(x, y) > k holds if and only if {u, v} = {x, y}. In particular, if the integer k is equal to the diameter of graph G, we get the goal-minimally k-diametric (k-GMD) graphs. In this paper we construct some infinite families of GME graphs and explore k-GME and k-GMD properties of cages.
This research was supported by the Slovak Scientific Grant Agency VEGA No. 1/0406/09. 相似文献
16.
Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p and p′ are joined by an edge if there is an element in G of order pp′. We denote by k(Γ(G)) the number of isomorphism classes of finite groups H satisfying Γ(G) = Γ(H). Given a natural number r, a finite group G is called r-recognizable by prime graph if k(Γ(G)) = r. In Shen et al. (Sib. Math. J. 51(2):244–254, 2010), it is proved that if p is an odd prime, then B p (3) is recognizable by element orders. In this paper as the main result, we show that if G is a finite group such that Γ(G) = Γ(B p (3)), where p > 3 is an odd prime, then \({G\cong B_p(3)}\) or C p (3). Also if Γ(G) = Γ(B 3(3)), then \({G\cong B_3(3), C_3(3), D_4(3)}\), or \({G/O_2(G)\cong {\rm Aut}(^2B_2(8))}\). As a corollary, the main result of the above paper is obtained. 相似文献
17.
Let ν(G) be the number of conjugacy classes of non-normal subgroups of a finite group G. The finite groups for which ν(G) ≤ 2 were determined by Dedekind and by Schmidt in the early times of group theory. On the other hand, if G is a finite p-group, La Haye and Rhemtulla have proved that either ν(G) ≤ 1 or ν(G) ≥ p. In this note, we determine all finite p-groups satisfying ν(G) = p for p > 2. 相似文献
18.
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. 相似文献
19.
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. 相似文献
20.
Letp>q and letG=Sp(p, q). LetP=LN be the maximal parabolic subgroup ofG with Levi subgroupL≅GL
q
(ℍ)×Sp(p−q). Forsεℂ andμ a highest weight of Sp(p−q), let пs,μ be the representation ofP such that its restriction toN is trivial and
⊠T
p-q
μ
, where det
q
is the determinant character of GL
q
(ℍ) andT
p-q
μ
is the irreducible representation of Sp(p−q) with highest weightμ. LetI
p,q(s, μ) be the Harish-Chandra module of the induced representation Ind
P
G
. In this paper, we shall determine the module structure and unitarity ofI
p, q(s, μ).
Partially supported by NUS grant R-146-000-026-112. 相似文献