# multiplication on permutation group written in cyclic notation

Sorry for stupid question but I didn't find any good explanation how to perform multiplication on permutation group written in cyclic notation.

For example, $a=(1352)$, $b=(256)$, $c=(1634)$, $ab=(1356)$, $ac=(1652)(34)$.

Why?

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In what order do you compose permutations? – lhf Apr 8 '11 at 16:31
This question may be of help, if it is not a duplicate in some sense already =) – Adrián Barquero Apr 8 '11 at 16:41

## 4 Answers

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 $1$: 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 $1$, which closes the cycle we have; 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 $2$. That is, we have $$ab = (1,3,5,2)(2,5,6) = (1,3,5,6)(2).$$ And finally we check what happens $4$: $(2,5,6)$ fixes $4$, as does $(1,3,5,2)$, so $4$ is fixed. So we have: $$ab = (1,3,5,2)(2,5,6)=(1,3,5,6)(2)(4) = (1,3,5,6).$$

Similarly 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.

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great explanation!!! Thanks! – com Apr 8 '11 at 17:00

The method I use for multiplying permutations like this is to think of each cycle as a set of mappings. a (in your example) maps 1 to 3, 3 to 5, 5 to 2, and 2 to 1. Also, remember that ab means "apply b, then apply a." So, here, we want to see where ab maps each number 1-6.

Start with 1: b fixes 1 (maps it to itself) and a maps 1 to 3. So we can begin writing ab = (13...

Now do 3: b fixes 3, and a maps 3 to 5. Put a 5 in: ab = (135...

Now 5: b maps 5 to 6 and a fixes 6, so ab = (1356...

Now 6: b maps 6 to 2 and a maps 2 to 1, so ab = (13561... = (1356).

Notice that ab fixes 4 since both a and b fix 4, but ab actually also fixes 2. This is because b maps 2 to 5, and a maps 5 right back to 2.

Hopefully you can use this method to check the other products.

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very helpful, thank you! – com Apr 8 '11 at 17:01

There is a small example on this page . Basically multiplication of permutation groups is applying permutations from right to left on an unaltered sequence.

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I just implement the permutation (you can also write a computer program to emulate these operations).

The priority of the product of permutation is from left to right. First write an origin array: 1 2 3 4 5 6 .... array(1)

Second use the first permutation (2 5 6) that is move array(1)'s 2nd element to 5th in new array as array(2) and etc. And we get a new array: 1 6 3 4 2 5 array(2)

Like the second step, we can get array: 6 2 1 4 3 5 as array(3)

Find the least element in array(3) whose position does not equal the value. It is 1. And find the number whose value is 1's position in array(3) (i.e. in this case the value is 3). And find the number whose value is 3's position in array(3) (i.e. in this case the value is 5). And do this repeatedly until find the number 1. Then list all these positions 3 5 6 1. This is actual permutation of the product of a and b.

Finally check that all the number in array(3) whose position does not equal the value are presented in the permutation. If not, find the minimal number which is not included in the list and whose position does not equal the value.

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