My professor has on our intranet uploaded some of his handwritten notes. I am a little worried about one of his statements and I'm suspecting that it contains a mistake.
"The dihedral group of degree 4 contains 4 different sylow 3-subgroups"
How is that possible?
The dihedral group of degree 4 ($D_{4}$) has order 8 and therefore every subgroup of $D_{4}$ need to have an order that divides 8 (according to lagrange) but a Sylow 3-subgroups has the order $3^{n}$ for some $1\leq n$, which makes the above statement false?