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Let $G = \{e, x, x^2, x^3, y, xy, x^2y, x^3y\}$ with $O(x)=4, O(y)=2$, and $xy=yx^3$. Then what is the number of elements in the centre of $G$?

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    O($x$) refers to the order of element x. – 2017-02-04
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    You may refer to the proof for [$Z(S_4) = \{e\}$](https://proofwiki.org/wiki/Center_of_Symmetric_Group_is_Trivial). – 2017-02-04
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    @GNU Supporter I think the given group is another group. – 2017-02-04
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    @MichaelRozenberg Thx for your comment. I confused $Z(S_4)$ with [$Z(D_4$)](https://ysharifi.wordpress.com/2011/02/02/center-of-dihedral-groups/). – 2017-02-04

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Sketch answer:

From the relation you can check that each $x^iy$ does not commute with $x$.

Next you can check if $x,x^3$ commute with $y$. You can check that what is left is the center of $G$.