I would really appreciate a step by step about how to solve the following:
C2 rwr C2 (where rwr is the regular wreath product)
i know it becomes (C2 x C2) ⋊ C2, so how to go from here.
edit: with solve i mean find the presentation of C2 rwr C2.
I would really appreciate a step by step about how to solve the following:
C2 rwr C2 (where rwr is the regular wreath product)
i know it becomes (C2 x C2) ⋊ C2, so how to go from here.
edit: with solve i mean find the presentation of C2 rwr C2.
So for the base group $C_2 \times C_2$ you have the presentation $\langle X,Y \mid X^2=Y^2=1, XY=YX \rangle$, which you correctly wrote down.
For the complement $C_2$, you should choose another letter for the generator, so make it $\langle Z \mid Z^2 = 1\rangle$.
To get the presentation of the wreath product, we combine these and use generators $X,Y,Z$ with all the relations you have already. But we need some extra relations which specify the action by conjugation of $C_2$ on $C_2 \times C_2$. Since this action is uniquely specified by the actions of generators on generators, you need just two further relations, which will be of the form
$Z^{-1}XZ = w_1$ and $Z^{-1}YZ = w_2$,
where $w_1$ and $w_2$ are elements of $C_2 \times C_2$, which you need to specify as words in the generators $X,Y$. I have left you to figure out what $w_1$ and $w_2$ should be.