Let $G$ be a group. An automorphism $\alpha$ of $G$ is called a commuting automorphism if $[\alpha(x), x] = 1$ for all $x \in G$. Let $A(G)$ be the set of all commuting automorphisms of $G$. A group $G$ is said to be an $A(G)$-group if $A(G)$ forms a subgroup of $\mathit{Aut}(G)$. We give some sufficient conditions on a finite $p$-group $G$ such that $G$ is an $A(G)$-group. As an application we prove that a finite $p$-group $G$ of coclass 2 for an odd prime $p$ is an $A(G)$-group. Also we classify non-$A(G)$ groups $G$ of order $p^{5}$.

For any abelian variety $A$ over a number field, we construct an extension of the Tate-Shafarevich group by the Bloch-Tamagawa space using the recent work of Lichtenbaum and Flach. This gives a new example of a Zagier sequence for the Selmer group of $A$.

Middle convolution is an operation for Fuchsian systems of differential equations which preserves Schlesinger’s deformation equations. In this paper we announce that Bolibruch’s non-Schlesinger deformations of Fuchsian systems are, in general, not preserved by middle convolution.

This paper presents new characterization for a non-compact Hermitian symmetric space $G/K$ to be of tube type (or non-tube type) by multiplicities in some branching laws and visible actions. The study in this paper gives an example of a kind of the Cartan decomposition for non-symmetric homogeneous spaces.