If a is the only element of order 2 then order of group is even
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Hi! mate GM here is your answer....,
Consider the element z=y−1xyz=y−1xy, we have: z2=(y−1xy)2=(y−1xy)(y−1xy)=ez2=(y−1xy)2=(y−1xy)(y−1xy)=e. So: z=xz=x, and y−1xy=xy−1xy=x. So: xy=yxxy=yx. So: xx is in the center of G.
I hope it will help you.
Consider the element z=y−1xyz=y−1xy, we have: z2=(y−1xy)2=(y−1xy)(y−1xy)=ez2=(y−1xy)2=(y−1xy)(y−1xy)=e. So: z=xz=x, and y−1xy=xy−1xy=x. So: xy=yxxy=yx. So: xx is in the center of G.
I hope it will help you.
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