#explain these examples | permutation groups | group theory

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dark scarab
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i think i just dont understand why n>=3 is necessary and needed, idk how were utilizing it in the example, ig for the contradiction but i still wanna confirm why is it identity only for n>=3. also i dont understand the group Z(Sn) i dont understand that group or notation and i dont wanna google it.

iin 4.14 i dont understand by beta how is 1 mentioned twice in two cycles and why is it 14523 and not 12345. also what is that final answer, why is it different after taking inverse.

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icy yarrow
# dark scarab i think i just dont understand why n>=3 is necessary and needed, idk how were u...

n>=3 Is needed cause you need a,b,c to be distinct, hence need at least 3 elements.

Z(G) is everything that commutes with everything in G (called the center of G)

1 is mentioned twice cause that's what beta is defined to be. 5 goes to 1 by the right cycle, then 1 goes to 2 by the left, so beta(5)=2

Just compute beta(i) for each i and see you get that cycle and not (12345)

Cause inverses usually don't equal the original permutation, and you can cycle the elements in a cycle. The just rewrote the permutation starting with 1. Again, direct computation verifies those cycles are equal

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I can go into more details on any in like 30-40 minutes

dark scarab
icy yarrow
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13254 does the same thing, they just started with where 1 goes

dark scarab
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Perfect thank u so much

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icy yarrow
icy yarrow