Exercise 2.1.4 ($H$ is closed under the group operation but is not a subgroup of $G$)

Question

Give an explicit example of a group $G$ and an infinite subset $H$ of $G$ that is closed under the group operation but is not a subgroup of $G$.

Richard Ganaye · Accepted Answer

ewcommand{\R}{\mathbb{R}}

\begin{proof} 
Consider the set $\R^+ \subset \R$ of positive numbers. Then for all real numbers $x,y$,
$$(x \in \R^+ 	ext{ and } y \in \R^+) \Rightarrow x + y \in \R^+,$$
but $\R^+$ is not a subgroup of the additive group $\R$, since $2 \in \R^+$, but $-2 
ot \in \R^+$.
\end{proof}

Exercise 2.1.4 ($H$ is closed under the group operation but is not a subgroup of $G$)

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Exercise 2.1.4 ($H$ is closed under the group operation but is not a subgroup of $G$)

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