Group Theory - Sylow Theory and simple groups

Quote:

Originally Posted by Jason Bourne

so the class equation is some thing like $|G| = |Z(G)| + \sum_{i=1}^{r} [G:N(x_i)]$

I don't think I understand the Class Equation properly. So I don't have this proof anywhere, $G/Z(G)$ is cyclic so this implies G is abelian, how?

Hi Jason Bourne.

Here is a detailed proof of the class equation posted here not so long ago: http://www.mathhelpforum.com/math-he...al-center.html

To show that $G/Z(G)$ cylic, say $G/Z(G)=\left<gZ(G)\right>$ for some $g\in G,$ implies $G$ Abelian, let $x,y\in G.$ Then $x,y$ belong to some left cosets of $Z(G)$ say $x=g^iz_1,\,y=g^jz^2$ for some $z_1,z_2\in Z(G)$ and $i,j\in\mathbb Z.$ It is then a straightforward matter to verify that $xy=yx.$