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Construct a group that has exactly 5 elements of order 4.

I wonder if it is possible. I tried $U(8)$ but it has $\{[1], [3], [5],[7]\}$ as elements which has order $4$ but it has only $4$ elements. So I am quite stuck.

Kara Smith
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1 Answers1

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An element has the same order as its inverse.

The inverse operation pairs off elements, except for those which are their own inverse.

Elements which are their own inverse have order 1 or 2.

Conclusion: elements of order 4 (or any other order greater than 2) come in pairs. It's not possible to have exactly 5.