How Do Permutations Like (a, b)(b, c) Result in (a, b, c)?

[cmurphy]

Why does (a, b)(b, c) = (a, b, c)

and why does (a, b)(c, d) = (a, b, c)(b, c, d). I don't understand how we get that c goes to b, since there is no d in the first cycle.

Colleen

 

[matt grime]

i think you need to learn what the notation means and how to use it, in the other thread i gave you an example of how to read things off. please note that when cycles are not disjoint all kinds of weird things can happen.

(12)(23)=(123) is a result of what the notation means.

imagine you've three balls in a line, red green blue in order.

doing (12)(23) tells you first to swap the balls in positions 2 and 3, so they look like
red blue green in order, and then two swap the *new* first and second position balls, so you get

blue red green

that is the same as doing

(123) which tells you to put the first ball in hte second position, the second ball in the 3rd position and the 3rd ball where hte 1st ball was.

try it: get two sets of three things that are distinctive, two apples, two oranges two bananas say and line em up, and do the permutations and see that they really are the same once you understand how to read the notation

 

[M98Ranger]

, the reason (a, b)(b, c) = (a, b, c) is because in a permutation, the elements within each cycle are not affected by the other cycles. In this case, (a, b) is one cycle and (b, c) is another cycle. The first cycle (a, b) simply switches the positions of a and b, while the second cycle (b, c) switches the positions of b and c. So when we combine these two cycles, we get (a, b)(b, c) = (a, c), because a stays in its original position and b and c switch.

Similarly, (a, b)(c, d) = (a, b, c)(b, c, d) because in this case, we have two separate cycles (a, b) and (c, d) that are not connected. So when we combine them, we get (a, b)(c, d) = (a, b, c)(b, d) where a and b stay in their original positions and c and d switch.

I hope this helps to clarify the concept of permutations and how combining cycles works. It is important to remember that in a permutation, the elements within each cycle are not affected by the other cycles.