Group of Order 4: Cyclic & C2xC2

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In summary, the conversation revolved around finding a condition that only concerns the elements of a group of order 4, specifically the two possible groups: the cyclic group and the product of two copies of the cyclic group. The initial suggestion was to use the fact that there are only two such groups to fill in the blank, but the goal was to find a more specific condition. The conversation also explored the idea of imposing equations, but it was noted that this would not work because C2 and C1 would also satisfy any equation satisfied by C4 or C2xC2. Finally, the conversation questioned what other types of conditions could be used, such as the elements forming a set of size 4, but it was unclear what the
  • #1
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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  • #2
If you need to fill it out then it's probably homework, right? Well, what have tried to do? What kind of 'condition'?
 
  • #3
i know there are only 2 such groups. the cyclic and the C2xC2
What's wrong with using that to fill in the blank?
 
  • #4
G is the unit group of either Z/5 or Z/8.
 
  • #5
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?
 
  • #6
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?
 

1. What is a group of order 4?

A group of order 4 is a mathematical concept that refers to a collection of 4 elements that follow certain rules and operations. In this case, we are specifically talking about a group with 4 elements that have the property of being cyclic or being the direct product of two cyclic groups.

2. What does it mean for a group to be cyclic?

A cyclic group is a group in which all elements can be generated by repeatedly applying a single element, known as a generator, to itself. In other words, the generator can "cycle" through all the elements of the group by multiplication. In a group of order 4, there will be one generator that can generate all 4 elements.

3. What is the difference between a group of order 4 and a group of order 2x2?

A group of order 2x2 refers to a group that is the direct product of two cyclic groups of order 2. This means that the group has 2 elements in each of its cyclic subgroups, resulting in a total of 4 elements. A group of order 4, on the other hand, can be cyclic or the direct product of two cyclic groups, but it does not necessarily have to be the direct product of two groups of order 2.

4. How many different groups of order 4 are there?

There are two different groups of order 4: the cyclic group of order 4 and the direct product of two cyclic groups of order 2. These two groups are isomorphic, meaning they have the same structure and can be mapped onto each other.

5. Can a group of order 4 have any other properties?

Yes, a group of order 4 can have other properties such as being abelian (commutative) or non-abelian (non-commutative). It can also have subgroups of different orders, depending on its structure. However, the two main properties we are discussing here are being cyclic or the direct product of two cyclic groups.

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