# Normal subgroups of finite index in infinite direct sum

As you may know, in a finite direct product $\prod_{i\in I}S_i$ of finite nonabelian simple groups $S_i$, every normal subgroup is of the form $\prod_{i\in J,J\subseteq I}S_i$.

Normal subgroups of finite index in infinite product $\prod_{i\in I}S_i$ (with $I$ infinite) are still of the same form?

Now we consider the direct sum $\oplus_{i\in I}S_i$ with $I$ infinite. The question is the same.

What are normal subgroups of finite index in the infinite sum $\oplus_{i\in I}S_i$?

Calling $\mathop{Supp} x = \{i\in I: x_i \ne 1\}$ the support of $x \in$ your infinite product/sum, take a look at the set $N_U$of all $x$ whose support is not an element of some fixed non-principal ultrafilter U on I. It's not too hard to show that $N_U$ is a normal subgroup. If all $S_i$ are isomorphic to each other, I think its index should be finite. –  j.p. Feb 27 '13 at 13:31