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Existence of normal Hall subgroups by means of orders of products
Author(s) -
Beltrán Antonio,
Sáez Azahara
Publication year - 2019
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201800165
Subject(s) - mathematics , sylow theorems , normal subgroup , order (exchange) , finite group , prime (order theory) , group (periodic table) , characterization (materials science) , pure mathematics , combinatorics , chemistry , materials science , organic chemistry , finance , economics , nanotechnology
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][A. Moretó, ]. As a consequence, we also give a π‐decomposability criterion for a finite group also by means of the orders of products.