Thread: 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

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 . That is the non-abelian group which is not or .
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.