2026/07/29 by Sonakshee Arora, Rahul Dattatraya Kitture
Mathematics · #math.GR #math.RT #msc:20C15 #msc:20E45 #msc:20E32
18 pages
arxiv created 2026/07/29 · arxiv updated 2026/07/30
Study of the structure of groups by variation in the arithmetic conditions on conjugacy classes and character degrees has produced several interesting results and open problems. In continuation of such work, Dolfi and Lucido in \citeMR1826493 introduced a property for groups. For primes p,q, a group G is said to have property P(p,q) if every p'-element in G has q'-class size. They obtained several results on the structure of G and of some subgroups when G satisfies the property P(p,q). Motivated by this work, we introduce a vanishing analogue of the above property: for primes p ≠ q, a finite group G is said to have the property Pv(p,q) if every vanishing p'-element of prime power order in G has conjugacy class size not divisible by q. We show that no finite simple group satisfies the property Pv(p,q) for primes p≠ q dividing |G|. We use this result to show that if a finite group G satisfies the property Pv(p,q) with p ≠ q and p > 2, then Oq'(G) (subgroup generated by all Sylow q-subgroups of G) is solvable. This generalises a result of Dolfi and Lucido under weaker conditions.