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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 $$\frac{|L|}{|G|} \leq a_k ?$$

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    $\begingroup$ 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). $\endgroup$ – Derek Holt Dec 12 '14 at 13:40
  • $\begingroup$ @DerekHolt this is great! Do you know what happens if we allow $G$ to be any finite group and not just a $p$-group? $\endgroup$ – Pablo Dec 12 '14 at 14:45
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    $\begingroup$ 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. $\endgroup$ – Derek Holt Dec 12 '14 at 15:58
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Problem 1. Classify the nonabelian p-groups G such that Z(G) is contained in the union of minimal nonabelian subgroups of G.

Problem 2 (old problem). Classify the p-groups covered by their minimal nonabelian subgroups.

A set of subgroups of a group G cover G is the union of these subgroups coincides with G. For example, the dihedral group D of order 2^4 is not covered by its minimal nonabelian subgroups since exp(D)=2^3 and exponents of all its minimal nonabelian subgroups equal 2^2.

Yakov

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    $\begingroup$ I don't see how these questions constitute an answer to Pablo's question. $\endgroup$ – Gerry Myerson Apr 3 '16 at 22:50

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