On the group with the same orders of Sylow normalizers as the finite projective special unitary group

In this paper we prove that a finite group G is isomorphic to the finite projective special unitary group Un(q) if and only if they have the same order of Sylow r-normalizer for every prime r.     

Solvability of any group with Hall supplements to normalizers of Sylow subgroups

Mathematical Notes - The solvability of any finite group with Hall supplements to normalizers of the Sylow subgroups is established.     

Sylow normalizers and character tables, II

Suppose thatG is a finitep-solvable group and letPε Syl _p (G). In this note, we prove that the character table ofG determines ifN _G(itP)/P is abelian. Research partially supported by DGICYT.     

Finite soluble groups with supersoluble Sylow normalizers

The authors are grateful to the Mathematical Institute of the University of Freiburg where part of the work was done for the kind hospitality and to P. Hauck for many helpful conversations.     

Symmetric and general linear groups with supersoluble Sylow normalizers

Sunto Nel presente lavoro vengono caratterizzati i gruppi finiti che hanno i normalizzanti dei sottogruppi di Sylow supersolubili e che appartengono alle seguenti classi: gruppi simmetrici, alterni, lineari generali e lineari speciali.     

Finite groups in which Sylow normalizers have nilpotent Hall supplements

The normalizer of each Sylow subgroup of a finite group G has a nilpotent Hall supplement in G if and only if G is soluble and every tri-primary Hall subgroup H (if exists) of G satisfies either of the following two statements: (i) H has a nilpotent bi-primary Hall subgroup; (ii) Let π(H) = {p, q, r}. Then there exist Sylow p-, q-, r-subgroups H _p , H _q , and H _r of H such that H _q ? N _H (H _p ), H _r ? N _H (H _q ), and H _p ? N _H (H _r ).     

On the power graph and the reduced power graph of a finite group

In this paper, for a finite group, we investigate to what extent its directed (resp. undirected) reduced power graph determines its directed power graph (resp. reduced power graph). Moreover, we investigate the determination of the orders of the elements of a finite group from its directed (resp. undirected) reduced power graph. Consequently, we show that some classes of finite groups are recognizable from their undirected reduced power graphs. Also, we study the relationship between the isomorphism classes of groups corresponding to the equivalence relations induced by the isomorphism of each of these graphs on the set of all finite groups.     

On the full automorphism group of a graph

While it is easy to characterize the graphs on which a given transitive permutation groupG acts, it is very difficult to characterize the graphsX with Aut (X)=G. We prove here that for the certain transitive permutation groups a simple necessary condition is also sufficient. As a corollary we find that, whenG is ap-group with no homomorphism ontoZ _p wrZ _p , almost all Cayley graphs ofG have automorphism group isomorphic toG.     

On the intersection graph of a finite group

For a finite group G, the intersection graph of G which is denoted by Γ(G) is an undirected graph such that its vertices are all nontrivial proper subgroups of G and two distinct vertices H and K are adjacent when H ∩ K ≠ 1. In this paper we classify all finite groups whose intersection graphs are regular. Also, we find some results on the intersection graphs of simple groups and finally we study the structure of Aut(Γ(G)).     

群与图的对称性

首先对平面图形的对称进行分析和利用其对称群进行量化,进而将此推广到考察一般图的广义"对称性"与图自同构群的关系,最后刻画了无平方因子阶局部本原弧传递图的自同构群结构.     

On the normalizers of the unitary subgroup in an integral group ring

In this paper, we investigate the properties of the normalizers of the unitary subgroup u_f(ZG) in an integral group rings. One of our main results is Theorem 2.6 which character¬izes the second normalizer of the unitary subgroup. As a conse¬quence of this theorem, we prove that the second normalizer of u_f(ZG) coincides with the first normalizer when G is a periodic group. Among other results, we give necessary and sufficient conditions for which the unitary subgroup is normal in the unit group when G is periodic and also characterize when all bicyclic units are nontrivial and elements of the normalizer of the unitary subgroup.