- #1
catcherintherye
- 48
- 0
I was wondering if anyone knows a more efficient method of finding conjugacy classes than the one i am currently using.
tex/ Example D_6* =<x,y| x^3=1, y^4=1, yx=x^2y>
now to find the conjugacy classes of this group i would first write out
the orbit of x <x> ={ 1x1, xxx^2, x^2xx, yxy^3,...x^2y^2xy^2x^2,...etc}
then i would use the set relation yx=(x^2)y to work out each of these 12 conjugates individually. Once this is done i continue with <x^2>, <y> etc...
..the only short cut i have found is the theorem that says <x>intersection<y> = empty set or <x>=<y>. But even with this surely there is a quicker way??
tex/ Example D_6* =<x,y| x^3=1, y^4=1, yx=x^2y>
now to find the conjugacy classes of this group i would first write out
the orbit of x <x> ={ 1x1, xxx^2, x^2xx, yxy^3,...x^2y^2xy^2x^2,...etc}
then i would use the set relation yx=(x^2)y to work out each of these 12 conjugates individually. Once this is done i continue with <x^2>, <y> etc...
..the only short cut i have found is the theorem that says <x>intersection<y> = empty set or <x>=<y>. But even with this surely there is a quicker way??