Question
Determine all subgroups of $\mathbf{R}^{*}$ (nonzero reals under multiplication) of index 2 .
Step 1
The index of a subgroup H in a group G, denoted by [G : H], is the number of distinct left cosets of H in G. In this problem, we are asked to find all subgroups of ℝ* with index 2, meaning there are exactly two distinct left cosets of the subgroup in ℝ*. Show more…
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Let $C^{\prime}=\left\{a \in G:(a x)^{2}=(x a)^{2}\right.$ for every $\left.x \in G\right\}$. Prove that $C^{\prime}$ is a subgroup of $G$.Let $C^{\prime}=\left\{a \in G:(a x)^{2}=(x a)^{2}\right.$ for every $\left.x \in G\right\}$. Prove that $C^{\prime}$ is a subgroup of $G$.
SUBGROUPS
D
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