0000000000512593
AUTHOR
Azahara Sáez
Existence of normal Hall subgroups by means of orders of products
Let G be a finite group, let π be a set of primes and let p be a prime. We characterize the existence of a normal Hall π‐subgroup in G in terms of the order of products of certain elements of G. This theorem generalizes a characterization of A. Moretó and the second author by using the orders of products of elements for those groups having a normal Sylow p‐subgroup 6. As a consequence, we also give a π‐decomposability criterion for a finite group also by means of the orders of products.
Order of products of elements in finite groups
If G is a finite group, p is a prime, and x∈G, it is an interesting problem to place x in a convenient small (normal) subgroup of G, assuming some knowledge of the order of the products xy, for certain p‐elements y of G.