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1.
Yakov Berkovich 《Israel Journal of Mathematics》2010,180(1):371-412
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 Sp2{\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 ≅ ΣSp2{\Sigma _{{p^2}}} in G is a multiple of p. 相似文献
2.
Let G be a nonabelian group and associate a noncommuting graph ∇(G) with G as follows: The vertex set of ∇(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. Abdollahi et al. (J Algebra 298(2):468–492, 2006) put forward a conjecture called AAM’s Conjecture in as follows: If M is a finite nonabelian simple group and G is a group such that ∇(G) ≅ ∇(M), then G ≅ M. Even though this conjecture is well known to hold for all simple groups with nonconnected prime graphs and the alternating
group A
10 [see Darafsheh (Groups with the same non-commuting graph. Discrete Appl Math (2008) doi:), Wang and Shi (Commun Algebra 36(2):523–528, 2008)], it is still unknown for all simple groups with connected prime graphs
except A
10. In the present paper, we prove that this conjecture is also true for the projective special linear simple group L
4(9). The new method used in this paper also works well in the cases L
4(4), L
4(7), U
4(7), etc. 相似文献
3.
Let G be a finite nonabelian group, ℤG its associated integral group ring, and Δ(G) its augmentation ideal. For the semidihedral group and another nonabelian 2-group the problem of their augmentation ideals
and quotient groups Q
n
(G) = Δ
n
(G)/Δ
n+1(G) is deal with. An explicit basis for the augmentation ideal is obtained, so that the structure of its quotient groups can
be determined. 相似文献
4.
Thomas P. Wakefield 《Algebras and Representation Theory》2012,15(3):427-448
Let G denote a finite group and cd (G) the set of irreducible character degrees of G. Bertram Huppert conjectured that if H is a finite nonabelian simple group such that cd (G) = cd (H), then G ≅ H × A, where A is an abelian group. Huppert verified the conjecture for PSp4(q) when q = 3, 4, 5, or 7. In this paper, we extend Huppert’s results and verify the conjecture for PSp4(q) for all q. This demonstrates progress toward the goal of verifying the conjecture for all nonabelian simple groups of Lie type of rank
two. 相似文献
5.
Let G be a non-abelian group and associate a non-commuting graph ∇(G) with G as follows: the vertex set of ∇(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. In this short paper we prove that if G is a finite group with ∇(G) ≅ ∇(M), where M = L
2(q) (q = p
n
, p is a prime), then G ≅ M.
相似文献
6.
Thomas P. Wakefield 《Algebras and Representation Theory》2011,14(4):609-623
Let G be a finite group and cd(G) be the set of irreducible character degrees of G. Bertram Huppert conjectured that if H is a finite nonabelian simple group such that cd(G) = cd(H), then G ≅ H×A, where A is an abelian group. In this paper, we verify the conjecture for the twisted Ree groups 2
G
2(q
2) for q
2 = 32m + 1, m ≥ 1. The argument involves verifying five steps outlined by Huppert in his arguments establishing his conjecture for many
of the nonabelian simple groups. 相似文献
7.
Schur multiplicators of infinite pro-<Emphasis Type="Italic">p</Emphasis>-groups with finite coclass
Bettina Eick 《Israel Journal of Mathematics》2008,166(1):147-156
Let G be an infinite pro-p-group of finite coclass and let M(G) be its Schur multiplicator. For p > 2, we determine the isomorphism type of Hom(M(G), ℤp), where ℤp denotes the p-adic integers, and show that M(G) is infinite. For p = 2, we investigate the Schur multiplicators of the infinite pro-2-groups of small coclass and show that M(G) can be infinite, finite or even trivial. 相似文献
8.
Let G be a nonabelian group. We define the noncommuting graph ∇(G) of G as follows: its vertex set is G\Z(G), the set of non-central elements of G, and two different vertices x and y are joined by an edge if and only if x and y do not commute as elements of G, i.e., [x, y] ≠ 1. We prove that if L ∈ {L
4(7), L
4(11), L
4(13), L
4(16), L
4(17)} and G is a finite group such that ∇(G) ≅ ∇(L), then G ≅ L. 相似文献
9.
Let G be a finite nonabelian group, P ∈Sylp(G), and bcl(G) the largest length of conjugacy classes of G. In this short paper, we prove that
in general and |P/Op(G)| < bcl(G) in the case where P is abelian.
Received: 26 December 2004; revised: 26 January 2005 相似文献
10.
In representation theory of finite groups, one of the most important and interesting problems is that, for a p-block A of a finite group G where p is a prime, the numbers k(A) and ℓ(A) of irreducible ordinary and Brauer characters, respectively, of G in A are p-locally determined. We calculate k(A) and ℓ(A) for the cases where A is a full defect p-block of G, namely, a defect group P of A is a Sylow p-subgroup of G and P is a nonabelian metacyclic p-group M
n+1(p) of order p
n+1 and exponent p
n
for
n \geqslant 2{n \geqslant 2}, and where A is not necessarily a full defect p-block but its defect group P = M
n+1(p) is normal in G. The proof is independent of the classification of finite simple groups. 相似文献
11.
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
12.
W. O. Alltop 《Israel Journal of Mathematics》1976,23(1):31-38
ItH
i
is a finite non-abelianp-group with center of orderp, for 1≦j≦R, then the direct product of theH
i
does not occur as a normal subgroup contained in the Frattini subgroup of any finitep-group. If the Frattini subgroup Φ of a finitep-groupG is cyclic or elementary abelian of orderp
2, then the centralizer of Φ inG properly contains Φ. Non-embeddability properties of products of groups of order 16 are established. 相似文献
13.
Let M be a simple group whose order is less than 108. In this paper, we prove that if G is a finite group with the same order and degree pattern as M, then the following statements hold: (a) If M ≠ A
10, U
4(2), then G ≅ M; (b) If M = A
10, then G ≅ A
10 or J
2 × ℤ3; (c) If M = U
4(2), then G is isomorphic to a 2-Frobenius group or U
4(2). In particular, all simple groups whose orders are less than 108 but A
10 and U
4(2) are OD-characterizable. As a consequence of this result, we can give a positive answer to a conjecture put forward by
W. J. Shi and J. X. Bi in 1990 [Lecture Notes in Mathematics, Vol. 1456, 171–180].
相似文献
14.
Teresa Cortadellas 《代数通讯》2013,41(2):601-612
In this article, we first give a necessary and sufficient condition on a finite purely nonabelian p-group G for the group Aut c (G) of central automorphisms of G to be elementary Abelian. We then generalize our result to the homocyclic case. 相似文献
15.
In §2, we prove that if a 2-group G and all its nonabelian maximal sub-groups are two-generator, then G is either metacyclic or minimal non-abelian. In §3, we consider a similar question for p > 2. In §4 the 2-groups all of whose minimal nonabelian subgroups have order 16 and a cyclic subgroup of index 2, are classified.
It is proved, in §5, that if G is a nonmetacyclic two-generator 2-group and A, B, C are all its maximal subgroups with d(A) ≤ d(B) ≤ d(C), then d(C) = 3 and either d(A) = d(B) = 3 (this occurs if and only if G/G′ has no cyclic subgroup of index 2) or else d(A) = d(B) = 2. Some information on the last case is obtained in Theorem 5.3. 相似文献
16.
Zvonimir Janko 《Mathematische Zeitschrift》2008,258(3):629-635
We determine here up to isomorphism the structure of any finite nonabelian 2-group G in which every two distinct maximal abelian subgroups have cyclic intersection. We obtain five infinite classes of such 2-groups
(Theorem 1.1). This solves for p = 2 the problem Nr. 521 stated by Berkovich (in preparation). The more general problem Nr. 258 stated by Berkovich (in preparation)
about the structure of finite nonabelian p-groups G such that A ∩ B = Z(G) for every two distinct maximal abelian subgroups A and B is treated in Theorems 3.1 and 3.2. In Corollary 3.3 we get a new result for an arbitrary finite 2-group. As an application
of Theorems 3.1 and 3.2, we solve for p = 2 a problem of Heineken-Mann (Problem Nr. 169 stated in Berkovich, in preparation), classifying finite 2-groups G such that A/Z(G) is cyclic for each maximal abelian subgroup A (Theorem 4.1).
相似文献
17.
Lingli Wang 《Frontiers of Mathematics in China》2010,5(1):179-190
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. 相似文献
18.
Let G be a finite non-Abelian group. We define a graph Γ
G
; called the noncommuting graph of G; with a vertex set G − Z(G) such that two vertices x and y are adjacent if and only if xy ≠ yx: Abdollahi, Akbari, and Maimani put forward the following conjecture (the AAM conjecture): If S is a finite non-Abelian simple group and G is a group such that Γ
S
≅ Γ
G
; then S ≅ G: It is still unknown if this conjecture holds for all simple finite groups with connected prime graph except
\mathbbA10 {\mathbb{A}_{10}} , L
4(8), L
4(4), and U
4(4). In this paper, we prove that if
\mathbbA16 {\mathbb{A}_{16}} denotes the alternating group of degree 16; then, for any finite group G; the graph isomorphism
G\mathbbA16 @ GG {\Gamma_{{\mathbb{A}_{16}}}} \cong {\Gamma_G} implies that
\mathbbA16 @ G {\mathbb{A}_{16}} \cong G . 相似文献
19.
Lingli Wang 《代数通讯》2013,41(2):523-528
Let G be a nonabelian group and associate a noncommuting graph ?(G) with G as follows: The vertex set of ?(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. In 1987, Professor J. G. Thompson gave the following conjecture. Thompson's Conjecture. If G is a finite group with Z(G) = 1 and M is a nonabelian simple group satisfying N(G) = N(M), then G ? M, where N(G):={n ∈ ? | G has a conjugacy class of size n}. In 2006, A. Abdollahi, S. Akbari, and H. R. Maimani put forward a conjecture (AAM's conjecture) in Abdollahi et al. (2006) as follows. AAM's Conjecture. Let M be a finite nonabelian simple group and G a group such that ?(G) ? ? (M). Then G ? M. In this short article we prove that if G is a finite group with ?(G) ? ? (A 10), then G ? A 10, where A 10 is the alternating group of degree 10. 相似文献
20.
In the following,G denotes a finite group,r(G) the number of conjugacy classes ofG, β(G) the number of minimal normal subgroups ofG andα(G) the number of conjugate classes ofG not contained in the socleS(G). Let Φ
j
= {G|β(G) =r(G) −j}. In this paper, the family Φ11 is classified. In addition, from a simple inspection of the groups withr(G) =b conjugate classes that appear in ϒ
j
=1/11
Φ
j
, we obtain all finite groups satisfying one of the following conditions: (1)r(G) = 12; (2)r(G) = 13 andβ(G) > 1; …; (9)r(G) = 20 andβ(G) > 8; (10)r(G) =n andβ(G) =n −a with 1 ≦a ≦ 11, for each integern ≧ 21. Also, we obtain all finite groupsG with 13 ≦r(G) ≦ 20,β(G) ≦r(G) − 12, and satisfying one of the following conditions: (i) 0 ≦α(G) ≦ 4; (ii) 5 ≦α(G) ≦ 10 andS(G) solvable. 相似文献