arXiv subjects

E. Ingrosso

Publications and source records attributed to E. Ingrosso.

2 recordsLinked to original sources

arXiv2026

On the independence of permutation characters

Let $G$ be a finite group. For every subgroup $H\leq G$, let $\pi_H$ be the permutation character of the action of $G$ on the left cosets of $H$. We prove that the characters $\pi_H$, with $H$ running through representatives of the conjugacy classes of subgroups of $G$, are linearly independent if and only if $G$ is cyclic. In particular, no finite insoluble group has the property asked for in Kourovka Notebook Problem~11.9. The proof uses only the fixed-point formula for a coset action and an elementary triangular-matrix argument.

math.GR↗

arXiv2026

Hopficity of profinite completions of abelian groups

We determine exactly when the profinite completion of an arbitrary abelian group is topologically Hopfian. For an abelian group $A$, we prove that \[ \widehat A \text{ is topologically Hopfian} \quad\Longleftrightarrow\quad A/pA \text{ is finite for every prime }p. \] As a byproduct, we answer Problem 6.30 of the Kourovka Notebook in the negative: for pairwise distinct odd primes $q_i$, the group $\bigoplus_{i\geq1}\Z[1/q_i]$ is residually finite and Hopfian, whereas its profinite completion is not topologically Hopfian.

math.GR↗