Centerless CR-radical

Question

D.J.S. Robinson, A Course in the Theory of Groups, 2d edition, 3.3.17, p. 88, states and proves : "In any group $G$ there is a unique maximal normal centerless CR-subgroup." ("CR" means "completely reducible". Robinson, p. 85, defines a completely reducible group as a group that is the "direct product" of a possibly infinite family of simple subgroups and in 3.3.1, he proves that a group generated by normal simple subgroups is completely reducible. Note that he says "direct product" where Scott says "direct sum" and where Bourbaki says "somme restreinte".)

This unique maximal normal centerless CR-subgroup of $G$ is called the centerless CR-radical of $G$. It seems clear to me that the centerless CR-radical of $G$ is in fact the largest normal centerless CR-subgroup of $G$.

The proof given by Robinson for the existence of the centerless CR-radical uses Zorn's lemma, but, if I'm not wrong, it can be made independent of Zorn's lemma and even of the axiom of choice.

My question is : do you know a reference for a proof independent of the axiom of choice ? Thanks in advance for the answers.

Answer 1

In any normal centerless CR-subgroup $N$ of $G$, the simple direct factors of $N$ are clearly subnormal subgroups of $G$, and they are nonabelian because $Z(N)=1$.

Now let $K$ be the subgroup generated by all subnormal nonabelian simple subgroups of $G$. All such subgroups commute (that result is due to Wielandt, and is proved in a later chapter of Robinson's book, in 13.3.1 in my edition), so $K$ is the direct product of all of these subgroups, and then clearly $K$ is the maximal centerless CR-subgroup $N$ of $G$.

That argument doesn't appear to use Zorn's Lemma.