Question

#38
http://i.gyazo.com/46f5bc52baecd530ffb8cc50c6c02b79.png

Answer 1

if nobody answers the first question check this link out. looks like a guy figured out :
https://answers.yahoo.com/question/index?qid=20090921143403AA7tpJg

Answer 2

Suppose \(x^5=e\). If \(x^k=e\) for \(k=1,2,3,4\) then we get that \(\gcd(k,5)=1\) for \(k=1,2,3,4\) implies there exists \(m,n\) s.t. \(km+5n=1\) so that \(x^{km}x^{5n}=x\implies x^{km}=x\implies x=e\). Since we are saying that \(x\) is not the identity we have that \(5\) is the order of \(x\).

Now surely we have \((x^2)^5=e\), \((x^3)^5=e\), and \((x^4)^5=e\), and in general \((x^{2+l})^5=e, (x^{3+l})^5=e, (x^{4+l})^5=e\) and \(x^{5l}\) for \(l=0,1,2,3,4,5,...\) so the solutions come if \(4's\) in the finite case.

Answer 3

The only thing I can think they want for the infinite case is to note that there will be infinite solutions?

Answer 4

\[\left( \begin{array}{cc} a & a\\ a & a \end{array} \right)
\left( \begin{array}{cc} x_{11} & x_{12} \\ x_{21} & x_{22} \end{array} \right)
=\left( \begin{array}{cc} a & a\\ a & a \end{array} \right)\]\[x_{11}+x_{21}=1,x_{12}+x_{22}=1\]is all that is needed i think

Answer 5

Ahh thanks! \(\phi(5)=4\). From your proof, looks it is related the number of integers less than \(5\) that are relatively prime to \(5\).

so, in a finite group, is it always the case that number of nonidentity elements of order \(k\) equals \(\phi(k)\) ?

@zzr0ck3r

Answer 6

\[\left( \begin{array}{cc} a & a\\ a & a \end{array} \right)
\left( \begin{array}{cc} x_{11} & x_{12} \\ x_{21} & x_{22} \end{array} \right)
=\left( \begin{array}{cc} a & a\\ a & a \end{array} \right)\]\[x_{11}+x_{21}=1,x_{12}+x_{22}=1\]

thnks @misty1212 , i think that will do.
since all the elements of the identity matrix must be equal, we get
\[\left( \begin{array}{cc} a & a\\ a & a \end{array} \right)
\left( \begin{array}{cc} x & x \\ x & x \end{array} \right)
=\left( \begin{array}{cc} a & a\\ a & a \end{array} \right)\]\[x+x=1 \implies x=1/2\]
so the identity must be \(\left( \begin{array}{cc} 1/2 & 1/2\\ 1/2 & 1/2 \end{array} \right)\)