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Mathematics
General Math
Math Proof Training and Practice
(open) Little Fermat Proof
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[QUOTE="fishturtle1, post: 6381877, member: 606256"] Proof: Let ##C## be the set of all ##p## colored chain with ##a## colors. Let ##c \in C##. We denote ##c = (c_1, c_2, \dots, c_p)## to be a ##p## colored chain where ##c_i## is any of the ##a## colors. Define ##\tau = (1 2 3 \dots p)##. Then ##G = \langle \tau \rangle## acts on ##C## by ##\tau^k \cdot c = (c_{\tau^k1}, c_{\tau^k2}, \dots, c_{\tau^kp})##. Moreover, every element of the orbit of ##c## represents the same chain, just rotated. Hence, by Burnside's Lemma $$N = \frac{1}{\vert G\vert} \sum_{k, 0 \le k \le p-1} F(\tau^k)$$ where ##N## is the number of orbits and ##F(\tau^k)## is the number of chains fixed by ##\tau^k##. Observe, ##F(1) = a^p## and for ##k > 0## we have ##F(\tau^k) = a## (since ##p## is prime we counted the number of mono color chains). Hence, $$N = \frac1p(a^p + (p-1)a)$$ Rearranging and subtracting ##pa## on both sides gives $$pN - pa = a^p - a$$ which implies ##p \vert (a^p - a)##. [] Edit: I'm sorry I just read the guidelines that maybe this post was for high school? I just learned Burnside's lemma and thought this would be a good way to practice, but feel free to delete what I wrote if i should not have posted. [/QUOTE]
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Forums
Mathematics
General Math
Math Proof Training and Practice
(open) Little Fermat Proof
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