abstract algebra

dori1123 · #1 $Old$ September 28th, 2008, 06:49 PM

|x| = order of x

Assume |x|=n and |y|=m. Suppose that x and y commute: xy = yx. Prove that |xy| divides the least common multiple of m and n.

Can anyone help?

NonCommAlg · #2 $Old$ September 28th, 2008, 08:07 PM

Quote:

Originally Posted by dori1123 $View Post$

|x| = order of x
Assume |x|=n and |y|=m. Suppose that x and y commute: xy = yx. Prove that |xy| divides the least common multiple of m and n.

if $\text{lcm}(m,n)=\ell,$ then obviously $x^{\ell}=y^{\ell}=1.$ thus since $xy=yx,$ we have: $(xy)^{\ell}=x^{\ell}y^{\ell}=1.$ hence $|xy| \mid \ell. \ \ \Box$

Remark: here are three related questions: suppose G is an abelian group and $|x|=n, \ |y|=m.$

Q1: show that it's possible to have $|xy| \neq \text{lcm}(m,n).$ (by giving an example!)

Q2: what can we say about $|xy|$ if $\gcd(m,n)=1$ ?

and a much more interesting question:

Q3: show that G has an element of order $\text{lcm}(m,n).$

dori1123 · #3 $Old$ September 28th, 2008, 10:15 PM

Quote:

Originally Posted by NonCommAlg $View Post$

Remark: here are three related questions: suppose G is an abelian group and $|x|=n, \ |y|=m.$

Q1: show that it's possible to have $|xy| \neq \text{lcm}(m,n).$ (by giving an example!)

Please give a hint. Thank you.