I'm trying to prove that the socle of a periodic divisible abelian group J is a proper subgroup of J. I know that J is direct sum of quasicyclic groups, say
$$ J={\oplus}_{i\in I} P_i $$
and that the socle of J is
$$ Soc(J)={\oplus}_p J[p] $$
Now, I'm wondering if it is true that
$$J[p]={\oplus}_{i\in I} P_i[p]$$
and that
$$Soc(J)={\oplus}_p\big({\oplus}_{i\in I}P_i[p]\big) ={\oplus}_{i\in I}\big({\oplus}_p P_i[p]\big) $$
Thank you very much!
I take $G[p]$ to mean the subgroup of elements satisfying $x^p=\rm identity$ in an abelian group $G$.
Yes, $\displaystyle\left(\bigoplus G_i\right)[p]=\bigoplus G_i[p]$ holds; this is because an element in the direct sum $\displaystyle\bigoplus G_i$ satisfies the relation $x^p=\rm identity$ if and only if each coordinate also satisfies this relation. So, yes, we have
$${\rm soc}(J)\cong\bigoplus_p J[p]\cong\bigoplus_p \left(\bigoplus_{i\in I} Q_i\right)[p]=\bigoplus_p\bigoplus_{i\in I} Q_i[p]\cong\bigoplus_{i\in I}\left(\bigoplus_p Q_i[p]\right)\cong \bigoplus_{i\in I}{\rm soc}(Q_i)$$
when $\displaystyle J=\bigoplus_{i\in I}Q_i$ (let's avoid overusing the letter 'P') over some index set $I$.
However, this sort of classification/decomposition theory is not fully necessary. To prove that the socle of a nontrivial torsion abelian divisible group $G$ is proper, show the following:
