(1) Let G be a group and H ⊂ Z(G) a subgroup such that G/H is cyclic. Prove that G is abelian. Conclude that every group of order p2 (p a prime number) is abelian and that the centre of any non-abelian group of order p3 has p elements. (2) (?) Let H be a normal subgroup of a p-group G, H 6= {e}. Prove that H∩Z(G) 6= {e}. In particular, one obtains that any normal subgroup with p-elements is con...

J. Browkin defined in his recent paper (Math. Comp. 73 (2004), pp. 1031–1037) some new kinds of pseudoprimes, called Sylow p-pseudoprimes and elementary Abelian p-pseudoprimes. He gave examples of strong pseudoprimes to many bases which are not Sylow p-pseudoprime to two bases only, where p = 2 or 3. In this paper, in contrast to Browkin’s examples, we give facts and examples which are unfavora...

objective: the purpose of this study is to compare the executive functions children and adolescents who suffer from attention deficit hyperactivity disorder (adhd) with normal children. method: twenty children with adhd were compared to 19 healthy children terms of some executive functions using the computerized version of tower london, continuous performance test (cpt), and stroop color test. ...

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 ...

We show that if the principal p-block of a finite group G contains exactly 5 irreducible ordinary characters, then Sylow p-subgroup has order 5, 7 or is isomorphic to one non-abelian 2-groups 8.

Perfect isometries and isotypies are constructed for alternating groups between blocks with abelian defect groups and the Brauer correspondents of these blocks. These perfect isometries and isotypies satisfy additional compatibility conditions which imply that an extended Broué conjecture holds for the principal block of an almost simple group with an abelian Sylow psubgroup and a generalized F...