TITLE: A group of order 20
QUESTION [6 upvotes]: Let $G$ be a group of order $20$ and its class equation is given by $$1+4+5+5+5$$
My question is whether Sylow-$2$ subgroup of $G$ is normal or not.
By Sylow theorem we know that the Sylow-$2$ subgroup will be normal if and only if there is only one Sylow-$2$ subgroup inside $G$. Sylow theorem says that number Sylow-$2$ subgroup is $2k+1$ where $2k+1$ divides $5$. Only $1$ and $5$ are possible. How to decide what will be the case here.
Thank you!

REPLY [3 votes]: Suppose, by way of contradiction, that there is a normal $2$-Sylow subgroup $D$.
If $D$ is cyclic, then it has exactly one involution, and this would form then a conjugacy class made of one element, which does not show up in the class equation. (The "$1$" in the class equation is already taken by the identity.)
This argument actually shows that there is no normal subgroup of order $2$.
If $D$ is a a Klein four-group, it has exactly three involutions, which would then yield a conjugacy class  of order $1$ or $3$, again not there.