Which group is C0?

Question

This is likely a silly question, but I cannot find an answer on the internet. I used Sage-math to find the structure of a group presented as:

$$\langle x,y; xy=1 \rangle$$

It returned C0: which group is it?

EDIT Here is the code I used

sage: G.<x,y,z,t>=FreeGroup()                                                                        
sage: H=G.quotient([x*y,x*z,x*t])                                                                    
sage: H.structure_description()                                                                      
'C0'

Answer 1

I'm not sure what $\;C_0\;$ means, but that group of yours is $\;\Bbb Z\;$ , the infinite cyclic group (though you wrote it multiplicatively). Observe that from the unique relation given, we get

$$xy=1\implies y=x^{-1}\implies K:=\langle x, y\;|\; xy=1\rangle=\langle x,x^{-1}\rangle =\langle x\rangle\cong\Bbb Z$$