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Two questions about cycles (algebra)... 
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#1
Dec2512, 11:21 AM

P: 247

I have two questions:
1) For the example on the second page, I don't understand why they say [tex]\alpha\gamma\alpha^{1} = (\alpha1 \alpha3)(\alpha2 \alpha4 \alpha7)(\alpha5)(\alpha6)[/tex] instead of [tex]\alpha\gamma\alpha^{1} = (\alpha1\alpha^{1} \alpha3\alpha^{1})(\alpha2\alpha^{1} \alpha4\alpha^{1} \alpha7\alpha^{1})(\alpha5\alpha^{1})(\alpha6\alpha^{1})[/tex]. 2) For the tables at the top of the 2nd page, I don't know how they computed those numbers... Thanks in advance 


#2
Dec2512, 04:07 PM

Sci Advisor
P: 3,252

If [itex] \alpha,\ p,\ q [/itex] are cycles, It is true that [itex] \alpha (\ p \ q) \ \alpha^{1} =( \alpha \ p \ \alpha^{1})(\alpha \ q \ \alpha^{1}) [/itex] but this is not the content of the theorem. A cycle is not the same as the product of the individual symbols in the cycle. The cycle (1,2,3) is not equal to (1)(2)(3). (There are 24 = (4)(3)(2) different permutations that can be formed by taking 3 distinct numbers from the set of numbers {1,2,3,4}. However, each permutation such as (1,2,3) is one of 3 representations of the same cycle. (1,2,3) = (2,3,1) = (3,1,2) So there are 8 = 24/3 distinct cycles of length 3 ) 


#3
Dec2612, 04:23 AM

P: 428

In a nutshell, notice that alpha gamma alpha inverse takes alpha of 1 to alpha of 3. ;)



#4
Dec2612, 04:39 PM

P: 247

Two questions about cycles (algebra)...



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