On the local Frattini subgroup of a finite group
Abstract
Let G be a finite group and let π(G) be the set of prime divisors of |G|. For each p∈ π(G), choose a Sylow p-subgroup Gp of G. The subgroup Ψ(G)= ∏p∈ π(G) Φ(Gp)G is called the \textit{local Frattini subgroup} of G. Here, for J leqslant G, the symbol JG denotes the core of J in G. The subgroup Ψ(G) is a characteristic subgroup of G that is contained in Φ(G), the Frattini subgroup of G. We prove that Ψ(G) satisfies many properties similar to Φ(G). Let J be a normal p-subgroup of G with J leqslant Φ(G), and let S be a Sylow p-subgroup of G. John S. Rose proved that if J leqslant Z(S), then J leqslant Φ(S). So, in fact, J leqslant Φ(S)G leqslant Ψ(G). Kaplan, Lev, Mann, and Solomon found various other conditions that would ensure J is contained in Φ(S). In this article, we add a few more conditions to this list.
Related
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BibTeX
@misc{costanzo2026local,
title = {{On the local Frattini subgroup of a finite group}},
author = {David G. Costanzo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16821},
eprint = {2610.16821},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16821},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the local Frattini subgroup of a finite group
AU - David G. Costanzo
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16821
UR - https://arxiv.org/abs/2610.16821
AB - Let G be a finite group and let π(G) be the set of prime divisors of |G|. For each p∈ π(G), choose a Sylow p-subgroup G_p of G. The subgroup Ψ(G)= ∏_p∈ π(G) Φ(G_p)_G is called the \textit{local Frattini subgroup} of G. Here, for J leqslant G, the symbol J_G denotes the core of J in G. The subgroup Ψ(G) is a characteristic subgroup of G that is contained in Φ(G), the Frattini subgroup of G. We prove that Ψ(G) satisfies many properties similar to Φ(G). Let J be a normal p-subgroup of G with J leqslant Φ(G), and let S be a Sylow p-subgroup of G. John S. Rose proved that if J leqslant Z(S), then J leqslant Φ(S). So, in fact, J leqslant Φ(S)_G leqslant Ψ(G). Kaplan, Lev, Mann, and Solomon found various other conditions that would ensure J is contained in Φ(S). In this article, we add a few more conditions to this list.
ER -