I have been given groups of order 4 such as $\mathbb{Z}_5^\times,\mathbb{Z}_8^\times$ etc and have been asked to determine whether they are $C_4$ or $K_4$ and I am not sure how to do this. I thought maybe if they are cyclic then they are $C_4$ but is this enough. (We have not proved that all groups of order 4 are $C_4$ or $K_4$ so cannot assume if they aren't $C_4$ than they are $K_4$)
Groups of Order 4, C4 or K4
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abstract-algebra
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0There is only one cyclic group on four elements (there's only one cyclic group on $n$ elements for all $n$; you should be able to prove this easily), so if you find an element of order $4$ then you know your group is a $C_4$. Otherwise, all three non-identity elements must have order 2 (why?); you should be able to show that the product of any two is the third (why?), and that then offers an easy isomorphism to $K_4$ (in fact, several different isomorphisms). – 2017-01-25
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0(Note that this effectively amounts to a proof that there are only two groups of order 4 in its own right, but you don't need to state that explicitly to be able to argue this way.) – 2017-01-25