# Sylow 2-subgroups

• Oct 1st 2008, 07:18 PM
IthacaPride
Sylow 2-subgroups
I am trying to show that given a cyclic sylow 2-subgroup of a group of order 2^n is normal in G. I *think* this is true, but I am not sure and it makes my proof of a larger problem simpler if it were true. Anyone know if it is or isn't? I was trying to think of a quick example that would disprove it but couldn't. thanks in advance
• Oct 1st 2008, 07:26 PM
ThePerfectHacker
Quote:

Originally Posted by IthacaPride
I am trying to show that given a cyclic sylow 2-subgroup of a group of order 2^n is normal in G. I *think* this is true, but I am not sure and it makes my proof of a larger problem simpler if it were true. Anyone know if it is or isn't? I was trying to think of a quick example that would disprove it but couldn't. thanks in advance

It is not true for $T$. That is the non-abelian group which is not $A_4$ or $D_6$.
It has three cyclic Sylow 2-subgroups. Therefore they are not normal.

EDIT: I am confused now, you said "... subgroup of a group of order 2^n". If the group has order 2^n then the Sylow 2-subgroup is the entire group itself. So it is definitely normal. But I do not think you really meant this.