arXiv subjects
Publications and source records attributed to Chiara Nicotera.
We consider groups $G$ such that the set $[G,φ]=\{g^{-1}g^φ|g\in G\}$ is a subgroup for every automorphism $φ$ of $G$, and we prove that there exists such a group $G$ that is finite and nilpotent of class $n$ for every $n\in\mathbb N$. Then there exists an infinite nonnilpotent group with the above property and the conjecture 18.14 of $[5]$ is false.
We show that there is a positive constant $δ< 1$ such that the probability of satisfying either the $2$-Engel identity $[X_1, X_2, X_2] = 1$ or the metabelian identity $[[X_1, X_2], [X_3, X_4]] = 1$ in a finite group is either $1$ or at most $δ$.
We study groups having the property that every non-abelian subgroup is equal to its normalizer. This class of groups is closely related to an open problem posed by Berkovich. We give a full classification of finite groups having the above property. We also describe all infinite soluble groups in this class.
Locally finite groups having the property that every non-cyclic subgroup contains its centralizer are completely classified.
We study groups having the property that every non-abelian subgroup contains its centralizer. We describe various classes of infinite groups in this class, and address a problem of Berkovich regarding the classification of finite $p$-groups with the above property.
We study groups having the property that every non-cyclic subgroup contains its centralizer. The structure of nilpotent and supersolvable groups in this class is described. We also classify finite $p$-groups and finite simple groups with the above defined property.
We investigate some situation in which automorphisms of a group G are uniquely determined by their restrictions to a proper subgroup H. Much of the paper is devoted to studying under which additional hypotheses this property forces G to be nilpotent if H is. As an application we prove that certain countably infinite locally nilpotent groups have uncountably many (outer) automorphisms.