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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.

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    What do you mean precisely(!) by "solve" or "calculate"?2012-11-13
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    ah yeah i missed that. edited.2012-11-13
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    anyone who can help?2012-11-14
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    Start by writing down presentations of $C_2 \times C_2$, and of $C_2$. Can you do that?2012-11-15
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    for C2 = < X | X^2 = 1 > and the other C2 = < Y | Y^2 = 1 >, the presentation of C2 x C2 will be < X,Y | X^2 = 1, Y^2 = 1, XY = YX >2012-11-16

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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.

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    w1 = X^-1 and w2 = Y^-12012-11-16
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    No - don't forget that all of these generators have order 2, so $X^{-1}=X$. The action in the wreath product is to interchange the to $C_2$s, so you want $Z^{-1}XZ=Y$ and $Z^{-1}YZ=X$.2012-11-16
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    ah ok. thank a lot for your help.2012-11-17