Let R be a commutative integral domain with quotient field K and let P be a nonzero strongly prime ideal of R. We give several characterizations of such ideals. It is shown that (P : P) is a valuation domain with the unique maximal ideal P. We also study when P^{&minus1} is a ring. In fact, it is proved that P^{&minus1} = (P : P) if and only if P is not invertible. Furthermore, if P is invertib...

Lagrange’s theorem tells us that if G is a finite group and H ≤ G, then #(H) divides #(G). As we have seen, the converse to Lagrange’s theorem is false in general: if G is a finite group of order n and d divides n, then there need not exist a subgroup of G whose order is d. The Sylow theorems say that such a subgroup exists in one special but very important case: when d is the largest power of ...

A finite group is called simple when its only normal subgroups are the trivial subgroup and the whole group. For instance, a finite group of prime size is simple, since it in fact has no non-trivial proper subgroups at all (normal or not). A finite abelian group G not of prime size, is not simple: let p be a prime factor of #G, so G contains a subgroup of order p, which is a normal since G is a...

For a given prime p, what is the smallest integer n such that there exists a group of order p in which the set of commutators does not form a subgroup? In this paper we show that n = 6 for any odd prime and n = 7 for p = 2.

LetN be a normal subgroup of a groupG and let p be a prime. We prove that if the p-part of jx j is a constant for every prime-power order element x 2 N n Z.N /, then N is solvable and has normal p-complement.

‎in this paper we give a new condition for a sylow $p$-subgroup of a finite group to control transfer‎. ‎then it is deduced a characteri-zation of supersoluble groups that can be seen as a generalization of the well known result concerning the supersolubility of finite groups with cyclic sylow subgroups‎. moreover a condition for a normal embedding of a strongly closed $p$-subgroup is given‎. ...

Let Gp be a Sylow p-subgroup of the finite groupG and let Char G n (Gp) represent the set of degree n complex characters of Gp that are the restrictions of class functions on G. We construct a natural map ψG : [BG,BU(n)] → ∏ p||G| Char G n (Gp) and prove that ψG is a surjection for all finite groups G that do not contain a subgroup isomorphic to (Z/p) for any prime p. We show, furthermore, that...