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GemmaJ1988

Member

Registered: 2008-10-08

Posts: 39

commutitive group

i have the following question to solve:
suppose G is a group with the property

for all x in G, prove that for all

in G, G is commutitive.

so far ive got:

so

is this enough to show there commutative?

Kurre

Member

Registered: 2006-07-18

Posts: 280

Re: commutitive group

i have the following question to solve:
suppose G is a group with the property

for all x in G, prove that for all

in G, G is commutitive.

so far ive got:

so

is this enough to show there commutative?

You are wrong at the underlined equation. For distinct x and y we dont have x*y=e. Only x*x=e and y*y=e.

To prove that G is commutative, with two elements x and y start with x*y, and try to arrive at y*x.
edit: Altough I see now, that a direct way from xy to yx seems hard to find, theres another way to do it. Hint: Since G is a group, we know that G is closed under multiplication. Thus xy is in G and yx is in G.

Last edited by Kurre (2009-01-24 07:04:27)

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: commutitive group

The important part of x^2 = e is a little hidden.  This is saying that inverse of x is equal to x, for all x in the group.  Now expand on what (xy)^-1 is using the Shoes-socks theorem, and it should come out clear as day.

---

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

mathsmypassion

Member

Registered: 2008-12-01

Posts: 33

Re: commutitive group

(x1*x2)*(x1*x2) = e
x1*(x2*x1)*x2 =e  / *x2 (operate to the right)
x1*(x2*x1)*e =x2
Now do the same thing to the left with x1
x1* /   x1*(x2*x1)=x2
e*(x2*x1) =x1*x2 or
x2*x1 = x1*x2
Q.E.D.

luca-deltodesco

Member

Registered: 2006-05-05

Posts: 1,470

Re: commutitive group

That looks a little messy, i think it's easier to write:

(xy)(xy) = e  (since xy is also in G)
x(yx)y = e
x(yx)y² = ey
x(yx) = y
x²(yx) = xy
yx = xy

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The Beginning Of All Things To End.
The End Of All Things To Come.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: commutitive group

These computations are fine, but they hide the significance of the problem behind number crunching/trickery.

See what I mean?  It directly comes from the fact that the inverse of xy is xy that the group is commutative.  You need to remark on the fact that there are two ways to compute the inverse of xy to solve this problem, the above is the straightforward way to do it.

---

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

mathsmypassion

Member

Registered: 2008-12-01

Posts: 33

Re: commutitive group

Thanks luca-deltodensco. Much better the way you wrote it.
Ricky, your solution is very ellegant. The way gemma tried to solve it made me consider a different solution than yours.

GemmaJ1988

Member

Registered: 2008-10-08

Posts: 39

Re: commutitive group

Thankyou guys so much!
I see now where i was going drastically wrong it seems relatively simple now