What do $\Bbb N^*$ and $\Bbb Z(p^n)$ mean in this paper?

Question

There is a theorem:

in this paper: http://journals.cambridge.org/download.php?file=%2FJAZ%2FJAZ78_01%2FS1446788700015548a.pdf&code=2ffd5c5100675caf83c2e95bce65491e

But there is no explanation of notations. What is $\Bbb N^\star$ and $\Bbb Z(p^n)$? Are they $\Bbb N\cup\{0\}$ and $\Bbb Z_{p^n}$?

Answer 1

In abelian group theory, it's customary to denote by $\mathbb{Z}(p^\infty)$ the Prüfer $p$-group (the $p$-torsion part of $\mathbb{Q}/\mathbb{Z}$ is a realization thereof).

The subgroups of this group form a chain: $$ \mathbb{Z}(p^0)\subset\mathbb{Z}(p^1)\subset\mathbb{Z}(p^2)\subset \dots\subset\mathbb{Z}(p^n)\mathbb{Z}(p^{n+1})\subset\dots\subset \mathbb{Z}(p^\infty) $$ and the subgroup $\mathbb{Z}(p^n)$ is cyclic of order $p^n$ (for finite $n$).

The meaning of $\mathbb{N}^*$ should be explained by the clause such that $p^lB=0$ holds for a nonnegative $l<n$; this strongly suggests $$ \mathbb{N}^*=\{1,2,3,\dotsc\} $$

The notation $\mathbb{Z}(p^n)$ and $\mathbb{N}^*$ in the sense explained above is widely used in Fuchs' Abelian Groups cited in the references.

Answer 2

primary component means the subgroup of all elements whose order is a power of $p$ and yes $\mathbb Z(p^n)$ is $\mathbb Z$/$p^n$$\mathbb Z$

$N^*$ should be non negative integer.