Number of Sylow $2$-subgroups in dihedral group $D_{20}$

Question

By Sylow's theorem I know that the number of Sylow $2$-subgroups in the symmetry group of a regular $10$-gon $D_{20}$ is either $1$ or $5$. How do I exclude the possibility $1$?

Answer 1

As a general piece of advice: when given a concrete group and asked about its Sylow subgroups, work with the concrete group. Don't just apply the Sylow theorems as if this were some strange abstract group. It is not a bad start but if you know more about the group than its cardinality, you should have more tools in the box.

So in this case you know the Sylow 2-subgroups have order $4$, because $4|20$ but $8\nmid 20$. So what could a Sylow 2-subgroup look like? It must be isomorphic to $\mathbf C_4$ or to $\mathbf C_2\times\mathbf C_2$. Since there are no elements of orde $4$ only the last option remains. So we are looking for two commuting elements of order $2$. If you start from a reflection, the only things it commutes with are the reflection about the line that is orthogonal to it, and the reflection about the origin i.e. rotation over $180^\circ$. So the Sylow 2-subgroups can be found as pairs of orthogonal reflections, and there are 5 such pairs. So there are $5$ Sylow 2-subgroups.

(Another argument would first observe that such subgroup generated by the reflections about orthogonal lines is a Sylow-subgroup because it has order $4$, and then use the fact that any two Sylow subgroups are conjugate to obtain that all others must also be of this type.)

Answer 2

All you need to do to exclude the $1$ possibility is to find at least $2$ such subgroups. Hint: consider reflections.