Multiplicative groups avoiding a fixed group

2024/09/10 by Hannesson, Matthias, Martin, Greg
#11N25 #11N37 #11N45 #11N64 #20K01 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2409.06869

Abstract

We know that any finite abelian group G appears as a subgroup of infinitely many multiplicative groups ℤ_n^× (the abelian groups of size ϕ(n) that are the multiplicative groups of units in the rings ℤ/nℤ). It seems to be less well appeciated that G appears as a subgroup of almost all multiplicative groups ℤ_n^×. We exhibit an asymptotic formula for the counting function of those integers whose multiplicative group fails to contain a copy of G, for all finite abelian groups G (other than the trivial one-element group).

Related

• Counting multiplicative groups with prescribed subgroups
• The smallest invariant factor of the multiplicative group
• The least primary factor of the multiplicative group
• The multisubset sum problem for finite abelian groups
• Arithmétique des Groupes Abéliens Finis