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Characterization of the orbits of a group action as equivalence classes


Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.18

Let $H$ be a group acting on a set $A$. Prove that the relation on $A$ defined by $a \sim b$ if and only if $a = h \cdot b$ for some $h \in H$ is an equivalence relation. (For each $x \in A$, the equivalence class of $x$ under $\sim$ is called the orbit of $x$ under $H$.)


Solution: We need to show that the relation $\sim$ is reflexive, symmetric, and transitive.

(1) For all $x \in A$, we have $x = 1 \cdot x$ so that $x \sim x$.

(2) Suppose $x \sim y$; then we have $x = h \cdot y$ for some $h \in H$. Then $y = h^{-1} \cdot x$, so that $y \sim x$.

(3) Suppose $x \sim y$ and $y \sim z$. Then there exist $h,k \in H$ such that $x = h \cdot y$ and $y = k \cdot z$. Then $$x = h \cdot y = h \cdot (k \cdot z) = (hk) \cdot z,$$ so that $x \sim z$ since $hk\in H$ as $H$ is a subgroup.

Thus $\sim$ is an equivalence relation.

Linearity

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