Totally Zippin $p$-groups

Abstract:

If $G$ is a $p$-group of limit length $\lambda$, then it satisfies the $\lambda$-Zippin property provided that whenever $A/{p^\lambda }A \cong G \cong B/{p^\lambda }B$, every isomorphism between ${p^\lambda }A$ and ${p^\lambda }B$ extends to an isomorphism between $A$ and $B$. We show that if $G$ is almost balanced in a totally projective group, then $G$ does satisfy the $\lambda$-Zippin property. This leads to the existence of a great variety of $G$’s that are totally Zippin in the sense that $G/{p^\alpha }G$ satisfies the $\alpha$-Zippin property for all limit ordinals $\alpha \leqslant \lambda = {\text {length of }}G$. Hence totally Zippin $p$-groups need not be $S$-groups, although those of countable length turn out to be direct sums of countable groups.