Breaking points in centralizer lattices

In this note, we prove that the centralizer lattice $\mathfrak{C}(G)$ of a group G cannot be written as a union of two proper intervals. In particular, it follows that $\mathfrak{C}(G)$ has no breaking point. As an application, we show that the generalized quaternion 2-groups are not capable.     

A counterexample to the local–global principle of linear dependence for Abelian varieties

Let A be an Abelian variety defined over a number field k. Let P be a point in $A(k)$ and let X be a subgroup of $A(k)$. Gajda and Kowalski asked in 2002 whether it is true that the point P belongs to X if and only if the point $(P\mathrm{mod}\mathfrak{p})$ belongs to $(X\mathrm{mod}\mathfrak{p})$ for all but finitely many primes $\mathfrak{p}$ of k. We provide a counterexample.