Question

Check whether H={x∈ R^*|x =1or x is irrational}and K={x∈ R^*|x ≥1}are subgroups of (R*,.).

Answer 1

Let us check whether $H=\{x\in R^*\ |\ x =1\text{ or }x\text{ is irrational }\}$ is a subgroup of $(R^*,\cdot)$. Since $\sqrt{2}\in H$ and $\sqrt{2}\cdot\sqrt{2}=2\notin H$, we conclude that $H$ is not a subgroup of $(R^*,\cdot)$.

Let us check whether $K=\{x\in R^*\ |\ x ≥1\}$ is a subgroups of $(R^*,\cdot)$. Taking onto account that for $x=2\in K$ the inverse $x^{-1}=\frac{1}{2}\notin K,$ we conclude that $K$ is not a subgroup of $(R^*,\cdot)$.