a $p$-group $g$ is called a $mathcal{cac}$-$p$-group if $c_g(h)/h$ is ‎cyclic for every non-cyclic abelian subgroup $h$ in $g$ with $hnleq‎ ‎z(g)$‎. ‎in this paper‎, ‎we give a complete classification of‎ ‎finite $mathcal{cac}$-$p$-groups‎.

‎In this paper‎, ‎some algebraic and geometrical properties of the rank$-k$ numerical hulls of normal matrices are investigated‎. ‎A characterization of normal matrices whose rank$-1$ numerical hulls are equal to their numerical range is given‎. ‎Moreover‎, ‎using the extreme points of the numerical range‎, ‎the higher rank numerical hulls of matrices of the form $A_1 oplus i A_2$‎, ‎where $A_1...

Gaifman’s normal form theorem showed that every first order sentence of quantifier rank n is equivalent to a Boolean combination of “scattered local sentences”, where the local neighborhoods have radius at most 7n−1. This bound was improved by Lifsches and Shelah to 3 · 4n−1. We use Ehrenfeucht-Fräıssé type games with a “shrinking horizon” to get a spectrum of normal form theorems of the Gaifma...

We introduce a notion of amenable normal extension S unital ring R with finite approximation system F, encompassing the algebras over field Gromov and Elek, twisted crossed product by an group, tensor extension. It is shown that every Sylvester matrix rank function rk preserved has canonical to rk_F for S. In case extension, it also depends on continuously. When group action C*-algebra preservi...

groups of infinite rank in which every subgroup is either normal or self-normalizing are characterized in terms of their subgroups of infinite rank.