Asymmetric decompositions of abelian groups

A subsetA of an Abelian groupG is said to be asymmetric ifg+S⊄A for any elementg∈G and any infinite symmetric subsetS⊂G (S=−S). The minimal cardinality of a decomposition of the groupG into asymmetric sets is denoted by ν(G). for any Abelian groupG, the cardinal number ν(G is expressed via the following cardinal invariants: the free rank, the 2-rank, and the cardinality of the group. In particular, . Translated fromMatematicheskie Zametki, Vol. 66, No. 1, pp. 10–19, July, 1999.     

Finite dimensional representations and subgroup actions on homogeneous spaces

LetH be an ℝ-subgroup of a ℚ-algebraic groupG. We study the connection between the dynamics of the subgroup action ofH onG/G _ℤ and the representation-theoretic properties ofH being observable and epimorphic inG. We show that ifH is a ℚ-subgroup thenH is observable inG if and only if a certainH orbit is closed inG/G _ℤ; that ifH is epimorphic inG then the action ofH onG/G _ℤ is minimal, and that the converse holds whenH is a ℚ-subgroup ofG; and that ifH is a ℚ-subgroup ofG then the closure of the orbit underH of the identity coset image inG/G _ℤ is the orbit of the same point under the observable envelope ofH inG. Thus in subgroup actions on homogeneous spaces, closures of ‘rational orbits’ (orbits in which everything which can be defined over ℚ, is defined over ℚ) are always submanifolds.     

Über Eine Klasse Endlicher Absoluter Geometrien

LetG be a finite group which is generated by a subsetS of involutions satisfying the theorem of the three reflections: Ifa,b,x,y,z S, ab 1 and ifabx,aby,abz are involutions, thenxyz S. Assume thatS contains three elements which generate a four-group. ThenS is a class of conjugate elements ofG if and only ifG/Z(G) is a non-abelian simple group. Moreover,G/Z(G) is a nonabelian simple group ifG is not isomorphic to any PGL₂(n).     

On weakly τ-quasinormal subgroups of finite groups

Let G be a finite group and H a subgroup of G. We say that: (1) H is τ-quasinormal in G if H permutes with all Sylow subgroups Q of G such that (|Q|, |H|) = 1 and (|H|, |Q ^G |) ≠ 1; (2) H is weakly τ-quasinormal in G if G has a subnormal subgroup T such that HT = G and T ∩ H ≦ H _τG , where H _τG is the subgroup generated by all those subgroups of H which are τ-quasinormal in G. Our main result here is the following. Let ℱ be a saturated formation containing all supersoluble groups and let X ≦ E be normal subgroups of a group G such that G/E ∈ ℱ. Suppose that every non-cyclic Sylow subgroup P of X has a subgroup D such that 1 < |D| < |P| and every subgroup H of P with order |H| = |D| and every cyclic subgroup of P with order 4 (if |D| = 2 and P is non-Abelian) not having a supersoluble supplement in G is weakly τ-quasinormal in G. If X is either E or F* (E), then G ∈ ℱ.     

A new characterization of frobenius complements

LetH, G be finite groups such thatH acts onG and each non-trivial element ofH fixes at mostf elements ofG. It is shown that, ifG is sufficiently large, thenH has the structure of a Frobenius complement. This result depends on the classification of finite simple groups. We conclude that, ifG is a finite group andA ⊆G is any non-cyclic abelian subgroup, then the order ofG is bounded above in terms of the maximal order of a centralizerC _G(a) for 1≠a ∈A.     

On s-quasinormal and c-normal subgroups of a finite group

Let F be a saturated formation containing the class of supersolvable groups and let G be a finite group. The following theorems are shown： （1） G ∈ F if and only if there is a normal subgroup H such that G/H ∈ F and every maximal subgroup of all Sylow subgroups of H is either c-normal or s-quasinormally embedded in G; （2） G ∈F if and only if there is a soluble normal subgroup H such that G/H∈F and every maximal subgroup of all Sylow subgroups of F（H）, the Fitting subgroup of H, is either e-normally or s-quasinormally embedded in G.     

A fixed point theorem for a class of star-shaped sets inc 0

A subsetK ofc ₀ is coordinatewise star-shaped (c.s.s.) if there exists a center pointx ∈K such that fory ∈K andz ∈c ₀, ifz is coordinatewise betweenx andy thenz ∈K. We prove that a weakly compact c.s.s. subset ofc ₀ has the fixed point property for nonexpansive mappings and that a fixed point for such a mapping can be obtained in a constructive manner. Research of the first two authors was partially supported by NSF Grant MCS78-01344 and of the last author by MCS78-01501.     

On <Emphasis Type="Italic">S</Emphasis>-quasinormal and <Emphasis Type="Italic">c</Emphasis>-normal subgroups of a finite group

Let be a saturated formation containing the class of supersolvable groups and let G be a finite group. The following theorems are presented: (1) G ∈ if and only if there is a normal subgroup H such that G/H ∈ and every maximal subgroup of all Sylow subgroups of H is either c-normal or S-quasinormally embedded in G. (2) G ∈ if and only if there is a normal subgroup H such that G/H ∈ and every maximal subgroup of all Sylow subgroups of F*(H), the generalized Fitting subgroup of H, is either c-normal or S-quasinormally embedded in G. (3) G ∈ if and only if there is a normal subgroup H such that G/H ∈ and every cyclic subgroup of F*(H) of prime order or order 4 is either c-normal or S-quasinormally embedded in G. Supported by the Natural Science Foundation of China and the Natural Science Foundation of Guangxi Autonomous Region (No. 0249001). Corresponding author. Supported in part by the Natural Science Foundation of China (10571181), NSF of Guangdong Province (06023728) and ARF(GDEI).     

On the Structure of Graphs with Bounded Asteroidal Number

 A set A⊆V of the vertices of a graph G=(V,E) is an asteroidal set if for each vertex a∈A, the set A\{a} is contained in one component of G−N[a]. The maximum cardinality of an asteroidal set of G, denoted by an (G), is said to be the asteroidal number of G. We investigate structural properties of graphs of bounded asteroidal number. For every k≥1, an (G)≤k if and only if an (H)≤k for every minimal triangulation H of G. A dominating target is a set D of vertices such that D∪S is a dominating set of G for every set S such that G[D∪S] is connected. We show that every graph G has a dominating target with at most an (G) vertices. Finally, a connected graph G has a spanning tree T such that d _T (x,y)−d _G (x,y)≤3·|D|−1 for every pair x,y of vertices and every dominating target D of G. Received: July 3, 1998 Final version received: August 10, 1999     

On solubility and supersolubility of some classes of finite groups

Let G be a finite group and H a subgroup of G. Then H is said to be S-permutable in G if HP = PH for all Sylow subgroups P of G. Let HsG be the subgroup of H generated by all those subgroups of H which are S-permutable in G. Then we say that H is S-embedded in G if G has a normal subgroup T and an S-permutable subgroup C such that T ∩ H HsG and HT = C. Our main result is the following Theorem A. A group G is supersoluble if and only if for every non-cyclic Sylow subgroup P of the generalized Fitting subgrou...     

Finite groups with <Emphasis Type="Italic">S</Emphasis>-supplemented <Emphasis Type="Italic">p</Emphasis>-subgroups

Consider a finite group G. A subgroup is called S-quasinormal whenever it permutes with all Sylow subgroups of G. Denote by B _sG the largest S-quasinormal subgroup of G lying in B. A subgroup B is called S-supplemented in G whenever there is a subgroup T with G = BT and B∩T ≤ B _sG . A subgroup L of G is called a quaternionic subgroup whenever G has a section A/B isomorphic to the order 8 quaternion group such that L ≤ A and L ∩ B = 1. This article is devoted to proving the following theorem.     

On J. H. C. Whitehead's aspherical question I

A connected, finite two-dimensional CW-complex with fundamental group isomorphic toG is called a [G, 2] _f -complex. LetL⊲G be a normal subgroup ofG. L has weightk if and only ifk is the smallest integer such that there exists {l ₁,…,l _k}⊆L such thatL is the normal closure inG of {l ₁,…,l _k}. We prove that a [G, 2] _f -complexX may be embedded as a subcomplex of an aspherical complexY=X∪{e ₁ ² ,…,e _k ² } if and only ifG has a normal subgroupL of weightk such thatH=G/L is at most two-dimensional and defG=defH+k. Also, ifX is anon-aspherical [G, 2] _f -subcomplex of an aspherical 2-complex, then there exists a non-trivial superperfect normal subgroupP such thatG/P has cohomological dimension ≤2. In this case, any torsion inG must be inP.     

Somep-groups of Frobenius and extra-special type

Finitep-groupsG are considered having a normal subgroupH ≠ 1 with this property: ifx ∈G/H andz ∈H thenx is conjugate inG toxz. Some theory is developed, and reasonably complicated examples of classes 2 and 3 are constructed.     

Division algebras with a projective basis

Letk be any field andG a finite group. Given a cohomology class α∈H ²(G,k ^*), whereG acts trivially onk ^*, one constructs the twisted group algebrak ^αG. Unlike the group algebrakG, the twisted group algebra may be a division algebra (e.g. symbol algebras, whereG⋞Z _n×Z_n). This paper has two main results: First we prove that ifD=k ^α G is a division algebra central overk (equivalentyD has a projectivek-basis) thenG is nilpotent andG’ the commutator subgroup ofG, is cyclic. Next we show that unless char(k)=0 and , the division algebraD=k ^α G is a product of cyclic algebras. Furthermore, ifD _p is ap-primary factor ofD, thenD _p is a product of cyclic algebras where all but possibly one are symbol algebras. If char(k)=0 and , the same result holds forD _p, p odd. Ifp=2 we show thatD ₂ is a product of quaternion algebras with (possibly) a crossed product algebra (L/k,β), Gal(L/k)⋞Z ₂×Z₂n.     

Restricted sumsets and a conjecture of Lev

LetA, B, S be finite subsets of an abelian groupG. Suppose that the restricted sumsetC={α+b: α ∈A, b ∈B, and α − b ∉S} is nonempty and somec∈C can be written asa+b witha∈A andb∈B in at mostm ways. We show that ifG is torsion-free or elementary abelian, then |C|≥|A|+|B|−|S|−m. We also prove that |C|≥|A|+|B|−2|S|−m if the torsion subgroup ofG is cyclic. In the caseS={0} this provides an advance on a conjecture of Lev. This author is responsible for communications, and supported by the National Science Fund for Distinguished Young Scholars (No. 10425103) and the Key Program of NSF (No. 10331020) in China.     

A Degree Sum Condition Concerning the Connectivity and the Independence Number of a Graph

Let G be a graph and S ⊂ V(G). We denote by α(S) the maximum number of pairwise nonadjacent vertices in S. For x, y ∈ V(G), the local connectivity κ(x, y) is defined to be the maximum number of internally-disjoint paths connecting x and y in G. We define . In this paper, we show that if κ(S) ≥ 3 and for every independent set {x ₁, x ₂, x ₃, x ₄} ⊂ S, then G contains a cycle passing through S. This degree condition is sharp and this gives a new degree sum condition for a 3-connected graph to be hamiltonian.     

Conjecture of Li and Praeger concerning the isomorphisms of Cayley graphs of A5

LetG be a finite group, andS a subset ofG \ |1| withS =S ⁻¹. We useX = Cay(G,S) to denote the Cayley graph ofG with respect toS. We callS a Cl-subset ofG, if for any isomorphism Cay(G,S) ≈ Cay(G,T) there is an α∈ Aut(G) such thatS ^α =T. Assume that m is a positive integer.G is called anm-Cl-group if every subsetS ofG withS =S ⁻¹ and | S | ≤m is Cl. In this paper we prove that the alternating groupA ₅ is a 4-Cl-group, which was a conjecture posed by Li and Praeger.     

Optimal Subgroups and Applications to Nilpotent Elements

Let G be a reductive group acting on an affine variety X, let x ∈ X be a point whose G-orbit is not closed, and let S be a G-stable closed subvariety of X which meets the closure of the G-orbit of x but does not contain x. In this paper we study G. R. Kempf’s optimal class Ω _G (x; S) of cocharacters of G attached to the point x; in particular, we consider how this optimality transfers to subgroups of G. Suppose K is a G-completely reducible subgroup of G which fixes x, and let H = C _G (K)⁰. Our main result says that the H-orbit of x is also not closed, and the optimal class Ω _H (x; S) for H simply consists of the cocharacters in Ω _G (x; S) which evaluate in H. We apply this result in the case that G acts on its Lie algebra via the adjoint representation to obtain some new information about cocharacters associated with nilpotent elements in good characteristic.     

Large centralizers in finite solvable groups

In 1955 R. Brauer and K. A. Fowler showed that ifG is a group of even order >2, and the order |Z(G)| of the center ofG is odd, then there exists a strongly real) elementx∈G−Z whose centralizer satisfies|C _G(x)|>|G|^1/3. In Theorem 1 we show that every non-abeliansolvable groupG contains an elementx∈G−Z such that|C _G(x)|>[G:G′∩Z]^1/2 (and thus|C _G(x)|>|G|^1/3). We also note that if non-abelianG is either metabelian, nilpotent or (more generally) supersolvable, or anA-group, or any Frobenius group, then|C _G(x)|>|G|^1/2 for somex∈G−Z. In Theorem 2 we prove that every non-abelian groupG of orderp ^mqⁿ (p, q primes) contains a proper centralizer of order >|G|^1/2. Finally, in Theorem 3 we show that theaverage |C(x)|, x∈G, is ≧c|G| ^1/3 for metabelian groups, wherec is constant and the exponent 1/3 is best possible.