Narrow Results

Let $G$ be a non-abelian $d$-generator finite $p$-group of order $p^n$. Ellis and McDermott in 1996 proved that $|G\otimes G|\le p^{nd}$. In the present paper, we improve this upper bound and show that $|G\otimes G|\le p^{(n-1)d+2}$. Also the $p$-groups with derived subgroup of order $p$ which attain the bound are obtained. Among other results, we classify all finite $p$-groups of order $p^n$, for $n\le 7$, with $|G\otimes G|=p^{nm-1}$, where $\left|G^{ab}\right|=p^m$.

A $p$-group $G$ is called a $C{C}_t^s$-group if $|G^{\prime }/(G^{\prime }\cap N)|$ $\le p^s$ or $|N/(N\cap Z(G))|\le p^t$ for every normal subgroup $N$ in $G$. In this paper, we first investigate the properties of $C{C}_t^s$-groups, in particular, we give the upper bounds of the exponent of $G^{\prime }$ with $G^{\prime }\le Z(G)$ and the order of $G^{\prime }Z(G)/Z(G)$ for a $C{C}_t^s$-group $G$. Then, we try to describe the structure of $C{C}_1^1$-groups, and a necessary condition for a $p$-group to be a $C{C}_1^1$-group is given. We also give some elementary properties of $p$-groups with very small derived subgroups by using the properties of capable $p$-groups, which maybe could apply some other research of $p$-groups.

The norm N(G) of a group G is the intersection of the normalizes of all the subgroups of G. A group is called capable if it is a central factor group. In this paper, we give a necessary and sufficient condition for a capable group to satisfy N(G)=ζ(G), and then some sufficient conditions for a capable group with N(G)=ζ(G) are obtained. Furthermore, we discuss the norm of a nilpotent group with cyclic derived subgroup.