Disjoint cycles

Shadow · **Posted:** Mon, 27 Feb 2012 07:35:18 UTC

DgrayMan wrote:

ah because theyre disjoint. they can't have anything in common to begin with right? so B cannot be an inverse of A because that would mean A and B are not disjoint right?

Yes.

DgrayMan · **Joined:** Sat, 21 Jan 2012 03:59:22 UTC **Posts:** 182

is there an algebraic way to show this? I mean i understand why it's true but other than just saying it in a sentence is there something more fancy?

and also, does this hold true for (AB)^t = e, where A^t=e and B^t=e for some positive integer t.?

Shadow · **Posted:** Tue, 28 Feb 2012 02:32:28 UTC

DgrayMan wrote:

is there an algebraic way to show this? I mean i understand why it's true but other than just saying it in a sentence is there something more fancy?

and also, does this hold true for (AB)^t = e, where A^t=e and B^t=e for some positive integer t.?

Of course, again, since they commute, (AB)^t=A^tB^t

so $A^t=B^{-t}$ so $A=B^{-1}$ another contradiction.

outermeasure · **Posted:** Tue, 28 Feb 2012 05:23:45 UTC

DgrayMan wrote:

is there an algebraic way to show this? I mean i understand why it's true but other than just saying it in a sentence is there something more fancy?

and also, does this hold true for (AB)^t = e, where A^t=e and B^t=e for some positive integer t.?

The effect of

on

would be moving it to a_2

. So ...

Shadow · **Posted:** Tue, 28 Feb 2012 05:27:54 UTC

outermeasure wrote:

DgrayMan wrote:

is there an algebraic way to show this? I mean i understand why it's true but other than just saying it in a sentence is there something more fancy?

and also, does this hold true for (AB)^t = e, where A^t=e and B^t=e for some positive integer t.?

The effect of

on

would be moving it to a_2

. So ...

Ah, fantastic, a much better way to see things.

Also, to clarify my statement, I meant that assuming you were doing that for all t>0 integers, it will happen to be true for some of them that it's true and that won't imply that both are the identity.

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Disjoint cycles

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