chief factor of finite group

moont14263

My question is about the shaded area in the attachment?
How did the author get that all the elements of order p or 4 of L are contained in K? I mentioned the abstract but I do not think there is a need for that.

Help?

Attached Thumbnails

 

micromass

Take [itex]x\in L[/itex]. Then there is a g in G such that [itex]x\in T^g[/itex] and thus there is a y in T such that [itex]x=gyg^{-1}[/itex]. If x has order p or order 4, then so does y. But by the previous sentence, y is contained in K. Since K is normal, we have that also x is in K.

moont14263

The symbol [itex]L=\bigcup_{g \in G} T^{g}[/itex], does it mean the union of sets or [itex]L=<T^{g},g \in G>[/itex]and, if it the union of sets, then how did he gets that [itex]L[/itex] equals to that union?