Orders of subgroups of Infinite Profinite Groups

Question

This question is in some sense equivalent to my question here. A proof would answer that question in the case when the base field is perfect.

Let $G$ be a profinite group of cardinality $\kappa$, where $\kappa$ is an infinite cardinal. Is it the case that for every infinite cardinal $\mu \leq \kappa$ there's a subgroup $H \subset G$ of cardinality $\mu$? As a follow-up does there need to be a such a subgroup that's normal (I'm just curious about this)?

I don't have a good intuition for infinite group theory and I have an even worse understanding of profinite groups, so I have made little progress on this question.

Edit: It seems to me that for a group $G$, a subgroup $H \subset G$ and $g \in G \setminus H$ we have that $|\langle g,H\rangle|\leq \max\{\aleph_0,|H|\}.$ This would seem to imply that a transfinite induction argument would answer the question yes in the general case.

Answer 1

Let $G$ be an infinite group of cardinality $\kappa$. Then, over ZFC, I can show:

1. For each infinite cardinal $\lambda < \kappa$ there is a proper subgroup of cardinality $\lambda$
2. In general there is no proper subgroup of cardinality $\kappa$
3. If $G$ is profinite, there are proper subgroups of all cardinalities $\le \kappa$

Proof: i) Choose a subset $X\subseteq G$ of cardinality $\lambda$. Then $H := \langle X \rangle$ is a subgroup of cardinality $\lambda$; in particular, it's a proper subgroup.

ii) $\mathbb{Z}_{p^\infty}:= \{z \in \mathbb{C}^\times\mid \exists n>0: z^{p^n}=1\}$ ($p$ a prime) is countably infinite, but all proper subgroups are finite (Rotman, Introduction to the Theory of Groups, Theorem 10.13).

iii) In the profinite case, $G=\varprojlim_N G/N$ where $G/N$ is finite. Hence the cardinality of $N$ is $\kappa$.