Multiplicative groups avoiding a fixed group
Matthias Hannesson, Greg Martin

TL;DR
This paper provides an asymptotic formula for counting integers whose multiplicative groups do not contain a given finite abelian group as a subgroup, highlighting that such groups appear in almost all multiplicative groups.
Contribution
It establishes an asymptotic count for integers with multiplicative groups avoiding a fixed finite abelian subgroup, extending understanding of subgroup appearances in multiplicative groups.
Findings
Almost all multiplicative groups contain any fixed finite abelian group as a subgroup.
An explicit asymptotic formula for the count of integers with multiplicative groups avoiding a given subgroup.
The result applies to all finite abelian groups except the trivial one-element group.
Abstract
We know that any finite abelian group appears as a subgroup of infinitely many multiplicative groups (the abelian groups of size that are the multiplicative groups of units in the rings ). It seems to be less well appeciated that appears as a subgroup of almost all multiplicative groups . We exhibit an asymptotic formula for the counting function of those integers whose multiplicative group fails to contain a copy of , for all finite abelian groups (other than the trivial one-element group).
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Taxonomy
TopicsFinite Group Theory Research · advanced mathematical theories
