Group of Order 4: Cyclic & C2xC2

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SUMMARY

A group G of order 4 can only be classified as either a cyclic group (C4) or the direct product of two cyclic groups of order 2 (C2 x C2). The discussion emphasizes the need for a specific condition regarding the elements of the group that distinguishes between these two types. A proposed condition is that if all elements of G have an order of 2, then G is isomorphic to C2 x C2. Conversely, if there exists an element of order 4, then G is cyclic (C4).

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Kanfoosh
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i need to fill this out:
G is a group of order 4 IFF ___________

i know there are only 2 such groups. the cyclic and the C2xC2
but i need to formulate a condition about the group's elements that can satisfy only one of the two.
 
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If you need to fill it out then it's probably homework, right? Well, what have tried to do? What kind of 'condition'?
 
i know there are only 2 such groups. the cyclic and the C2xC2
What's wrong with using that to fill in the blank?
 
G is the unit group of either Z/5 or Z/8.
 
it's not a homework question, i just thought it's simpler to write it like that

i need a condition that concerns only the elements of the group. i.e. given a group G whose elements all are ________ then |G|=4
perhaps a preceding question should be: is there any such condition?
 
i need a condition that concerns only the elements of the group.
Could you be more explicit in what this means?

Obviously, simply imposing equations won't work. (because C2 and C1 will satisfy any equation satisfied by C4 or C2xC2)

But you seem to reject more general things, like the condidition that the elements form either C4 or C2xC2. And, I presume you'd reject the condition that "the elements form a set of size 4".

So just what sorts of conditions do you have in mind?
 

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