Multiplication in Permutation Groups Written in Cyclic Notation

I didn’t find any good explanation how to perform multiplication on permutation group written in cyclic notation. For example, if
a=(1352),b=(256),c=(1634),
then why does ab=(1356) and ac=(1652)(34)?

Answer

You are thinking of the permutations as functions, so when you write “ab“, you mean that you perform the permutation b first, and the permutation a second.

Here’s one way to do it: write the disjoint cycle expressions for both a and b, in the given order:
(1,3,5,2)(2,5,6)
Now, moving from right to left, see what happens to each number in each cycle.

For instance, start with 1, so we write 1 down:
(1,
The first cycle, (2,5,6), does nothing to 1, so it stays 1. Then the next cycle, (1,3,5,2), sends 1 to 3. So, in total, 1 is sent to 3. We write
(1,3,
Now consider 3. The first cycle, (2,5,6), does nothing to 3. The second maps 3 to 5. So the product maps 3 to 5. So now we have
(1,3,5,
Now 5. The first cycle, (2,5,6), sends 5 to 6; the second cycle does nothing to 6, so in total, 5 is sent to 6. So for the product we now have
(1,3,5,6,
Next, what happens to 6? The first cycle sends 6 to 2; and then the next cycle sends 2 to 1. So 6 is sent to 1, and because we started out with 1, this now closes the cycle we have; thus, we also close the bracket. So the product so far is
(1,3,5,6)
Now we consider the “next” number that hasn’t been described yet, 2. The first cycle, (2,5,6), sends 2 to 5; then we check what the next cycle does to 5, which is that it sends it back to 2. So 2 maps to 2, and since we started out with 2, it again closes the cycle. So now we have
(1,3,5,6)(2)
Finally we check what happens 4, as it’s the remaining number: (2,5,6) fixes 4 (it doesn’t do anything to it – it remains as it is), as does (1,3,5,2), so 4 is overall fixed. So now finally we have:
ab=(1,3,5,2)(2,5,6)=(1,3,5,6)(2)(4)=(1,3,5,6)
\therefore (1,3,5,2)(2,5,6)=(1,3,5,6)

It’s similar with ac. Here we have:
(1,3,5,2)(1,6,3,4).
First consider 1: the first cycle maps it to 6, the second cycle fixes 6. So 1\mapsto 6. Then 6 is sent to 3 by the first cycle, and 3 to 5 by the second cycle (reading right to left, remember), so 6\mapsto 5. Then 5 is fixed by the first cycle and sent to 2 by the second cycle, so 5\mapsto 2. Then 2 is fixed by the first cycle and sent to 1 by the second, which means 2\mapsto 1, closing the cycle: we have (1,6,5,2). The next number not already covered is 3; 3 is mapped to 4 by the first cycle (by b), and 4 is fixed by a, so 3\mapsto 4. Then 4 is sent to 1 by the first cycle, and 1 is sent to 3 by the second cycle, so this closes this second cycle as (3,4). Putting the two together we get
(1,3,5,2)(1,6,3,4) = (1,6,5,2)(3,4)
as given.

Attribution
Source : Link , Question Author : com , Answer Author : Skeleton Bow

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