Elementary Proof of a Theorem of Hawkes, Isaacs and Özaydin

2019/07/01 by van der Lee, Matthé
#05E15 (Secondary) #11A25 (Secondary) #20D30 (Primary) #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1907.00513

Abstract

We present an elementary proof of the theorem of Hawkes, Isaacs and Özaydin, which states that Σ μ_G(H,K)≡ 0 mod d, where μ_G denotes the Möbius function for the subgroup lattice of a finite group G, H ranges over the conjugates of a given subgroup F of G with [G:F] divisible by d, and K over the supergroups of the H for which [K:H] divides d. We apply the theorem to obtain a result on the number of solutions of |⟨ H,g⟩|| n, for said H and a natural number n. The present version of the article includes an additional result on a quantity studied by K.S. Brown.

Related

• On two Möbius function for a finite non-solvable group
• The Möbius function of the small Ree groups
• The Moebius function on the lattice of normal subgroups
• The A-Möbius function of a finite group
• A Note on the Solvablity of Groups