# Why $G/C_G(a) leq G/zeta(G)$ in this lemma?

Mathematics Asked by M.Ramana on February 24, 2021

My question:

Could you please tell me why $$G/C_G (a)leq G/zeta (G)$$ in the last sentence? Here $$zeta (G)$$ is the center of $$G$$. Thanks in advance.

I am sorry, but the answer of Chris Custer is not correct. You have $$zeta(G) subseteq C_G(a) subseteq G$$, and this only implies that $$G/C_G(a) cong (G/zeta(G))/(C_G(a)/zeta(G))$$. Probably you need divisibility and locally finiteness along the line to show that $$G/C_G(a)$$ is isomorphic to a direct summand of $$G/zeta(G)$$.

Note added later Aha! Now I see the full proof. So indeed the statement you are questioning is totally wrong. As I pointed out above, $$G/C_G(a)$$ is a quotient of a divisible group hence divisible. That is what is needed. And of course that a bounded divisible group is trivial. So thanks for adding the full proof and good question you asked! I already +1-ed your post.

Correct answer by Nicky Hekster on February 24, 2021