How to Calculate Permutation Powers with Functional Notation?

[vivaitalia1]

I might be a bit thick in this but i just can't figure out how to answers this exam question:

Calculate p to the power of 100, writing your answer in functional notation

p is the permutation(n=10): (3,5,7,6,2,9,1,10,8,4). It should say 1-10 on the top but i don't know how to draw matrices here.

Anyways, point is, how do you do these kinds of questions? :(

 

[Data]

Decompose it into a product of disjoint cycles. In this case you'll find p = (1, 3, 7)(2, 5)(4, 6, 9, 8, 10).

The second cycle up there has order 2 and the last has order 5, so p^{100} = (1, 3, 7)^{100}.

Notice that (1, 3, 7)^2 = (1, 7, 3) and (1, 7, 3)^2 = (1, 3, 7). That should be enough for you to work it out.

 

[vivaitalia1]

So is the question infact asking me to run the permutation through 100 cycles?

 

[matt grime]

No, the question is asking yo to raise dsjoint 3, 2 and 5 cycles to the power 100. Clearly two of those 2 disjoint cycles raised to the power 100 are the identity, and the remaining on is just that cycle again. You understand that if a group element, g, has order n, then g^m is the same as g^r where r=m mod n, right?

 

[vivaitalia1]

Yes i know that. thanks a lot for the explanation :)

Just one last thing.. what's 'functional notation'? is it the disjoint cycles notation? so basically my answer would be the same as the dijoint cycle notation for this permutation?

i missed soo many lectures (im studying comp science) and basicalyl I am teaching myself the whole syllabus

 

[Data]

Functional notation probably means the type that you tried to use in your first post (the type that looks sort of like a matrix).

 

[Chris Hillman]

vivaitalia1, another thing you might think about is whether the author of your textbook is multiplying permutations left to right or right to left. Unfortunately, modern algebra books seem to be divided about fifty-fifty. (One author, Herstein, even wrote a classic modern algebra textbook using one convention and then wrote a second modern algebra textbook using the other convention!) This can cause great confusion to unwary but diligent students who are comparing treatments in different books! (Otherwise a smart thing to do.)