If G is a group, $H< G^{|G|}$ and $|H|\geq 2^{|G|}$, then can $H$ be cyclic?

Question

Let G be a group, $H< G^{|G|}$ and $|H|\geq 2^{|G|}$. Can $H$ be a cyclic group?

By $G^{|G|}$ I mean the direct product of $|G|$ copies of $G$.

I was wondering if someone could help me about the above question.

Answer 1

Set $n=|G|$. Let $(g_1,\ldots,g_n)$ be an element of $G^{n}$. Then it is of order $\leq n$ since $(g_1,\ldots,g_n)^n=(g_1^n,\ldots,g_n^n)=(e,\ldots, e)$. So any cyclic subgroup of $G^n$ has order $\leq n$. But $n<2^n$, so there is no $H$ as in your question.