Let G be a finite group in which every element has a square root

[robertjordan]

Homework Statement

Let G be a finite group in which every element has a square root. That is, for each x in G, there exists a y in G such that y^2=x. Prove every element in G has a unique square root.

Homework Equations

G being a group means it is a set with operation * satisfying:
1.) for all a,b,c in G, a*(b*c)=(a*b)*c
2.) there exists an e s.t. for all x in G, x*e=x
3.) for all x in G there exists an x' s.t. x'*x=e

The Attempt at a Solution

The only thing I have gotten so far is the reason why (-2)^2 and 2^2 both equaling 4 doesn't present a counterexample. The reason it doesn't is because (-2) has no square root so it is not in our group G.

... other than that I'm stuck..

 

[Dick]

You are probably overthinking this. If you have two different elements of the group, x1 and x2 and y1 is a square root of x1 and y2 is a square root of x2, then y1 is not equal to y2, yes? So you have at least as many square roots in G as there are elements of G. What happens if an element has two square roots? Use that G is finite.

 

[robertjordan]

You are probably overthinking this. If you have two different elements of the group, x1 and x2 and y1 is a square root of x1 and y2 is a square root of x2, then y1 is not equal to y2, yes? So you have at least as many square roots in G as there are elements of G. What happens if an element has two square root? Use the G is finite.

Every element in G has a square root. If an element has 2 square roots then two elements in G have the same square root (by the pidgeonhole principle). That means there exists a y in G s.t. y^2=x1 and y^2=x2 where x1=/=x2. This is a contradiction because it would suggest y^2=/=y^2.


Is this right?

Thanks for your help

 

[Dick]

Every element in G has a square root. If an element has 2 square roots then two elements in G have the same square root (by the pidgeonhole principle). That means there exists a y in G s.t. y^2=x1 and y^2=x2 where x1=/=x2. This is a contradiction because it would suggest y^2=/=y^2.


Is this right?

Thanks for your help

You've got the idea, but I guess I would start with proving all of square roots are distinct elements of G first, instead of leaving it for a contradiction at the end. And why does G have to be finite?

 

[robertjordan]

You've got the idea, but I guess I would start with proving all of square roots are distinct elements of G first, instead of leaving it for a contradiction at the end.

I don't get what you mean by this. Do you mean prove there is no element y such that y is the square root of x1 and x2, x1=/=x2? Or do you mean prove that for any x, the "square root of x" is well-defined?

I don't see what's wrong about what I did? I assumed BWOC that there existed an element with 2 different square roots and I showed how that led to a contradiction... Doesn't that imply no element has 2 square roots? And thus every element has a unique square root?


Thanks

And why does G have to be finite?

If G was infinite we couldn't use the pidgeonhole principle?

 

[Dick]

I mean prove there is no element y such that y is the square root of x1 and x2, x1=/=x2. And yes, I know it's obvious. "If an element has 2 square roots then two elements in G have the same square root", that doesn't ring right. In the integers mod 3, 1^1=1 and 2^2=1. So the element 1 has two square roots. That doesn't mean two elements have the same square root. At least until you explain it better. And, yes, I know 2 doesn't have a square root.

I'm not saying it's wrong. Just explain what the pigeons are and what the holes are.

 

[robertjordan]

I mean prove there is no element y such that y is the square root of x1 and x2, x1=/=x2. And yes, I know it's obvious. "If an element has 2 square roots then two elements in G have the same square root", that doesn't ring right. In the integers mod 3, 1^1=1 and 2^2=1. So the element 1 has two square roots. That doesn't mean two elements have the same square root. At least until you explain it better. And, yes, I know 2 doesn't have a square root.

I'm not saying it's wrong. Just explain what the pigeons are and what the holes are.

I should have said: "If an element has 2 square roots then two elements in G share a square root."

The reason I feel that's true is because since G is finite, there are only |G| possible options for an element's square root. And there are also |G| elements each of which has a square root. So there are |G| pidgeons (being the elements of G), and |G| holes (the possible square roots). Therefore if one pidgeon has 2 square roots, then that leaves |G|-2 pidgeonholes for |G|-1 remaining pidgeons... and thus two elements must share a square root.

x1 and x2 sharing a square root means there is a y in G such that y^2=x1 and y^2=x2... but that's a contradiction?Is that right?

 

[Dick]

I should have said: "If an element has 2 square roots then two elements in G share a square root."

The reason I feel that's true is because since G is finite, there are only |G| possible options for an element's square root. And there are also |G| elements each of which has a square root. So there are |G| pidgeons (being the elements of G), and |G| holes (the possible square roots). Therefore if one pidgeon has 2 square roots, then that leaves |G|-2 pidgeonholes for |G|-1 remaining pidgeons... and thus two elements must share a square root.

x1 and x2 sharing a square root means there is a y in G such that y^2=x1 and y^2=x2... but that's a contradiction?


Is that right?

Yes, it sounds correct.