# Subgroups of the union of conjugates

This question is an attempt to find a version of this question with a positive answer. In this question, we ask if one can find big subgroups in the union of conjugtes of small subgroups of finite $p$-groups.

Fix a prime number $p$. Is there a sequence $\{a_k\}_{k \in \mathbb{N}}$ of real numbers with $$\lim_{k \to \infty} a_k = 0$$

such that for any finite $p$-group $G$, any $k \in \mathbb{N},$ any subgroup $H \leq G$ with $[G : H] \geq k$, and any subgroup $L \leq G$ such that $$L \subseteq \bigcup_{g \in G} gHg^{-1}$$ we have that $$\frac{|L|}{|G|} \leq a_k ?$$

• I think this is true with $a_k=1/k$ i.e. $|L| \le |H|$ in this situation. It can be proved by induction on $|G|$. Choose a normal subgroup $N$ of $G$ with $|N|=p$, and apply induction to $G/N$ distinguishing between the cases $N \le H$ and $N \cap H=1$ (note that $L \cap N=1$ in the second case). – Derek Holt Dec 12 '14 at 13:40
• @DerekHolt this is great! Do you know what happens if we allow $G$ to be any finite group and not just a $p$-group? – Pablo Dec 12 '14 at 14:45
• I am really not sure about that - I would not like to bet either way on what happens for general finite solvable groups, for example. – Derek Holt Dec 12 '14 at 15:58