|G|=4. Prove the group is either cyclic or g^2=e

In summary: If x has order 2, then x^2=e and each element is its own inverse. If x has order 4, then x is a generator and the group is cyclic. Therefore, G is either cyclic or all elements have order 2.
  • #1
mathmajor2013
26
0
Let G be a group with |G|=4. Prove that either G is cyclic or for any x in G, x^2=e.
 
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  • #2
Consider and element x in G of maximum order. What possibilities are there, and what does each of them say about the group structure?
 
  • #3
So the order of x is 2 or 4, right? Since the order must divide 4. If it's 2, the x^2=e, and so each element is its own inverse. If it's 4...I'm lost.
 
  • #4
Correct. And if it's 4, what can you say about x, x^2,x^3 and x^4? Are any of them equal? What is x for this group?
 
  • #5
Damn. They are all unique, and x^4=e. and x is the generator for the group yeah?
 
  • #6
So the order of x is 2 or 4, right? Since the order must divide 4. If it's 2, the x^2=e, and so each element is its own inverse. If it's 4...I'm lost.
 
  • #7
You are correct. The order of each element in the group must divide the order of the group. So, the order of each element must be 1,2 or 4. If there are no elements of order 4, then x^2=e for each x in the group (since the order of each element is 1 or 2). If there is an element of order 4, then this element is a generator, as you pointed out, and so the group is cyclic.
 
  • #8
mathmajor2013 said:
Let G be a group with |G|=4. Prove that either G is cyclic or for any x in G, x^2=e.

a non identity element must have either order 4 or order 2
 

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

A cyclic group is a group that can be generated by a single element, meaning all other elements in the group can be expressed as powers of that element.

2. How can I tell if a group is cyclic?

A group is cyclic if there exists an element, called a generator, that can be used to generate all other elements in the group through repeated multiplication or exponentiation.

3. What is the significance of g^2=e in relation to the group being cyclic?

If g^2=e, where e is the identity element of the group, it means that g is its own inverse. This is a property of cyclic groups, as every element in a cyclic group has an inverse that is also a power of the generator.

4. How can I prove that a group is cyclic?

To prove a group is cyclic, you can show that there exists an element that can generate all other elements in the group. This can be done by demonstrating that every element in the group can be expressed as a power of the generator.

5. Can a group be both cyclic and have g^2=e?

Yes, a group can be both cyclic and have g^2=e. This is because g^2=e only means that g is its own inverse, which is a property of cyclic groups. However, not all groups with g^2=e are necessarily cyclic.

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