Can anybody help me in finding the all possible subgroups of the group
<x,y:x²=y³=(xy)¹²=1>. Actually i want to find the the number of all subgroups of this group.
Are you sure you meant the above or perhaps you meant to give a presentation of a group by generators and relators, in the
form of ?
Because if you meant what you wrote and not the above then we get:
, from where we get
that your group can be presented as ... Check this.
I suspect he didn't see the `=1' in your presentation, but even then your group abelianises to . So...yeah, he's done something wrong...I have not idea where he gets his final presentation from.
Anyway, to address your question, there isn't really a `neat' way of finding all the subgroups of a group that I know of. You just need to do it by hand. So, what is the order of your group? Which groups have order dividing your groups order?
Next, take elements from your group and see what (sub)groups they generate. Then you want to write the subgroups out in a lattice, much like you can find here.