Characterization of epicenter of finite abelian group

Statement

Version for an abelian group of prime power order

Suppose $G$ is an abelian group of prime power order with underlying prime number $p$ . The classification of finite abelian groups tells us that the possible abelian groups of order $p^{k}$ are given by the set of unordered integer partitions of $k$ . Explicitly, the group corresponding to the partition $k=m_{1}+m_{2}+\dots +m_{r}$ , with $m_{1}\geq m_{2}\geq \dots \geq m_{r}$ , is:

$\mathbb {Z} /p^{m_{1}}\mathbb {Z} \times \mathbb {Z} /p^{m_{2}}\mathbb {Z} \times \dots \mathbb {Z} /p^{m_{r}}\mathbb {Z}$

The characterization of the epicenter of $G$ , if given in the above form, is as follows:

If $r=1$ (i.e., $G$ is cyclic) then the epicenter is the whole group $G$ .
If $r\geq 2$ , the epicenter is (using additive notation for the abelian group $G$ ) $p^{m_{2}}G$ , i.e., all elements that are $p^{m_{2}}$ times an element of $G$ . Note that as an abstract group, this is isomorphic to $\mathbb {Z} /p^{m_{1}-m_{2}}\mathbb {Z}$ .

Observations

The epicenter is trivial (making $G$ a capable group) if and only if $m_{1}=m_{2}$ , i.e., there are two cyclic factors of order equal to the exponent of $G$ .
The epicenter itself is a cyclic group of prime power order.

Version for a finite abelian group

In terms of Sylow subgroups: We can first write the finite abelian group as an internal direct product of its Sylow subgroups. Then, compute the epicenter of each Sylow subgroup using the version for an abelian group of prime power order, and finally take the internal direct product back to get the epicenter of the whole group.

We can also give a direct description: if we consider the invariant factors of $G$ , the epicenter is the subgroup of $G$ obtained as follows:

If there is a single invariant factor, it is the whole group $G$ .
Otherwise, it is $dG$ where $d$ is the second last invariant factor (where the invariant factors are arranged in ascending order).

=Observations

The epicenter is trivial (making $G$ a capable group) if and only if the last two invariant factors are equal.
The epicenter itself is a finite cyclic group.

Facts

The following follow directly from the characterization: