I have a equation related Burnside's Lemma $|X/G|=\frac{1}{|G|}\sum_{g\in G}|X^g|$. Can anyone give me intuitive explanation of the equation?Since I'm a novice learner of Burnside's Lemma I need explanation of each term of the above equation.
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asked Dec 27, 2014 at 9:52
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1 Answer
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Here a group $G$ acts on a set $X$. $X/G$ is the quotient of $X$ by $G$; that is, the set of orbits $\{G\cdot x\mid x\in X\}$, or equivalently the set of equivalence classes of $X$ where the equivalence relation is that $x\sim y$ if there is an element $g\in G$ so that $g\cdot x=y$. $X^g$ is the set of fixed points of a group element $g\in G$; i.e. $X^g=\{x\in X\mid g\cdot x=x\}$. So what the equation is saying that the umber of orbits is equal to the average number of fixed points.
