Cosets and cyclic groups

[Kate2010]

Homework Statement

Consider the cyclic group C_n = <g> of order n and let H=<g^m> where m|n.

How many distinct H cosets are there? Describe these cosets explicitly.

Homework Equations

Lagrange's Theorem: |G| = |H| x number of distinct H cosets

The Attempt at a Solution

|G| = n
I'm unsure if I have interpreted H correctly, I think it is the cyclic group generated by g^m so contains g^mk for integers k. I don't know how I can work out the order of H from this, or how to describe the cosets.

 

[Martin Rattigan]

Fine so far. When will g^mk=g^mk' for different integers k, k'?

 

[Kate2010]

Would it be when k' is a multiple of k?

 

[Martin Rattigan]

No. If you have a cyclic group (g) of order 20, and H=(g²) then the values g^2x0, g^2x1, g^2x2, ... g^2x9 will all be different, then you start again. Here m is 2 and k goes from 0 to 9, but for instance 1 divides each value of k, and 2 divides 4, but g² is not the same as g⁴,g ⁶,g⁸ etc. and g^2x2=g⁴ is not the same as g^2x4=g⁸.

How would you get the powers of g in the equation g^mk=g^mk' to the same side of the equals sign? Then you can use gⁿ=e.

 

[Martin Rattigan]

Actually if you think about it, if g^mk=g^mk' were equivalent to k' is a multiple of k, it would also have to be equivalent to k is a multiple of k'.

This would mean the two values would be the same iff k=±k', but that would imply an infinite set ogf different values and there are only n.

 

[Kate2010]

Is it when k=n/m?

 

[Martin Rattigan]

Well I think it's getting closer. Would that work with the example I gave? n/m in that instance would be 10. But where does k' come into it?

We want to know how many different values (g^m)^k runs through as k runs through all the integers. It can be at most n, but will it be all n? Would (g²)^k give all 20 values in the example. Would you get g⁵ for some k for instance?

The idea of trying to find when two different k give the same result is that you would probably then see how many different groups of k give the same result. The size of H is then the number of groups. But if you look at the example carefully the result will probably become pretty obvious.

What for instance would be the next value of k in the example such that (g²)^k=(g²)³?