# Thread: subgroups of the group <x,y:x²=y³=(xy)¹²>

1. ## subgroups of the group <x,y:x²=y³=(xy)¹²>

Can anybody help me in finding the all possible subgroups of the group
<x,y:x²=y³=(xy)¹²>. Actually i want to find the the number of all subgroups of this group.

2. Originally Posted by makenqau
Can anybody help me in finding the all possible subgroups of the group
<x,y:x²=y³=(xy)¹²>. Actually i want to find the the number of all subgroups of this group.

$x^2=(xy)^{12}\Longrightarrow x=y(xy)^{11}\,,\,and\,\,also\,\,y^3=(xy)^{12}\Long rightarrow y^2=(xy)^{11}x=(xy)^{11}y(xy)^{11}$ , thus it

follows that $\langle x,y\;;\;x^2=y^3=(xy)^{12}\rangle =\langle {y,(xy)^{11}\rangle \cong F_2=$ the free group in two (free) generators.

Now, did you actually mean to write what you did or you meant to give a presentation of the group in the

form $\langle x,y\;;\;x^2,y^3,(xy)^{12}\rangle$ ? Because this last is NOT a free group as it has non-trivial torsion elements, so

we get two different groups...or I'm wrong, of course. Check this.

Tonio