Let G $G$ be a finite group admitting coprime automorphism α $\alpha$ . J ( ) $J_G(\alpha )$ denote the set of all commutators [ x , ] $[x,\alpha ]$ where $x$ belongs to an -invariant Sylow subgroup We show that $[G,\alpha is soluble or nilpotent if and only any generated by pair elements orders from nilpotent, respectively.

For a finite group G, subgroup P of G is 2-minimal if B<P, where B=NG(S) for some Sylow 2-subgroup S and B contained in unique maximal P. Here we give detailed explicit description all the subgroups symplectic groups defined over field odd characteristic.

We show that the character table of a finite group $G$ determines whether Sylow 2-subgroup is generated by 2 elements, in terms Galois action on characters. Our proof this result requires use Classification Finite Simple Groups and provides new evidence for so-far elusive Alperinâ??McKayâ??Navarro conjecture.

We examine the projective dimensions of Mackey functors and cohomological Mackey functors. We show over a field of characteristic p that cohomological Mackey functors are Gorenstein if and only if Sylow p-subgroups are cyclic or dihedral, and they have finite global dimension if and only if the group order is invertible or Sylow subgroups are cyclic of order 2. By contrast, we show that the onl...

A subgroup H is said to be s-permutable in a group G, if HP = PH holds for every Sylow subgroup P of G. If there exists a subgroup B of G such that HB = G and H permutes with every Sylow subgroup of B, then H is said to be SS-quasinormal in G. In this paper, we say that H is a weakly SS-quasinormal subgroup of G, if there is a normal subgroup T of G such that HT is s-permutable and H ∩ T is SS-...

Suppose that G is an abelian p-separable group and R is a commutative unital ring of prime characteristic p. It is proved that if Rp i possesses nilpotent elements for each natural number i, then the normalized Sylow p-subgroup S(RG) in the group ring RG is torsion-complete if and only if G is p-bounded. This supplies our recent results from Acta Math. Hung. (1997), Ric. Mat. (2002), Tamkang J....

Let G be a finite group and let T (G) be the abelian group of equivalence classes of endotrivial kG-modules, where k is an algebraically closed field of characteristic p. We determine, in terms of the structure of G, the kernel of the restriction map from T (G) to T (S), where S is a Sylow p-subgroup of G, in the case when S is abelian. This provides a classification of all torsion endotrivial ...

We define some new kinds of pseudoprimes to several bases, which generalize strong pseudoprimes. We call them Sylow p-pseudoprimes and elementary Abelian p-pseudoprimes. It turns out that every n < 1012, which is a strong pseudoprime to bases 2, 3 and 5, is not a Sylow p-pseudoprime to two of these bases for an appropriate prime p|n− 1. We also give examples of strong pseudoprimes to many bases...