On the number of p -elements in a finite group
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On the number of p‑elements in a finite group Pietro Gheri1 Received: 8 July 2020 / Accepted: 28 August 2020 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, we study the ratio between the number of p-elements and the order of a Sylow p-subgroup of a finite group G. As well known, this ratio is a positive integer and we conjecture that, for every group G, it is at least the (1 − p1 )-th power of the number of Sylow p-subgroups of G. We prove this conjecture if G is p-solvable. Moreover, we prove that the conjecture is true in its generality if a somewhat similar condition holds for every almost simple group. Keywords Finite groups · p-Elements · Sylow subgroups · Subnormalizers · p-Frobenius ratio Mathematics Subject Classification 20D20 · 20D35
1 Introduction Let G be a finite group and p be a prime dividing the order of G. Moreover, let ⋃ 𝔘p (G) = P, P∈Sylp (G)
be the set of p-elements of G. A celebrated theorem of F.G. Frobenius ([3]) states that if P is a Sylow p-subgroup of G, then |P| divides |𝔘p (G)| . We will call the positive integer |𝔘p (G)|∕|P| the p-Frobenius ratio of G. The number of p-elements of a finite group is a fundamental invariant in finite group theory. Several different proofs of Frobenius’ theorem have been given (see, for
This work is dedicated to the memory of Carlo Casolo. His knowledge, his curiosity, his humility and his humanity were an example to all of his students and friends. * Pietro Gheri [email protected] 1
Dipartimento di Matematica e Informatica “U. Dini”, Università degli Studi di Firenze, viale Morgagni 67/a, 50134 Florence, Italy
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example, [6, 10]). Moreover in [11][Theorem 15.2] it is proven that the p-Frobenius ratio in a finite group of Lie type is equal to the size of a Sylow p-subgroup. Nevertheless, it is still unknown if the Frobenius ratio has a combinatorial meaning. It is clear that the p-Frobenius ratio is 1 if and only if G contains a normal Sylow p-subgroup. In [8][page 80], it is proven with a nice and easy argument that if the p-Frobenius ratio is not 1, then it must be greater or equal than p. In this paper, we focus on the search for “good” bounds for the p-Frobenius ratio in terms of the number np (G) of Sylow p-subgroups of G. Of course, a trivial upper bound is obtained when every pair of Sylow p-subgroups of G has trivial intersection, so that, given a Sylow p-subgroup P of G,
|𝔘p (G)| |P|
≤ np (G) −
np (G) − 1 |P|
≤ np (G).
It is not hard to find examples of sequences of groups that show that a lower bound on the p-Frobenius ratio cannot be linear in np (G) . We state the following conjecture. Conjecture A Let G be a finite group, p be a prime dividing |G| and P a Sylow p-subgroup of G. Then
|𝔘p (G)| |P|
≥ np (G)
1− p1
(1)
.
We will show in Example 2.5 that this bound is “asymptotically tight”. We show that Conjecture A is true for p-solvable groups. Namely, we prove the following. Theorem A Let G be a fi
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