Question
Show, by example, that in a factor group $G / H$ it can happen that $a H=b H$ but $|a| \neq|b|$.
Step 1
Given a group G and a normal subgroup H, the factor group G/H is the set of all left cosets of H in G, with the group operation defined as (aH)(bH) = (ab)H. Now, let's consider an example. Let G = Z_6 (the integers modulo 6) and H = {0, 3}, which is a normal Show more…
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Let $G$ be an abelian group. If $H=\left\{x \in G: x=x^{-1}\right\}$, that is, $H$ consists of all the elements of $G$ which are their own inverses, prove that $H$ is a subgroup of $G$.
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