For some reason I'm having trouble with this simple exercise. I'm sure I'm simply overlooking something. Thanks.
Show that $\displaystyle \{x \in D_n | x^2=e\}$ is not a subgroup of $\displaystyle D_n (n\geq 3) $.
For some reason I'm having trouble with this simple exercise. I'm sure I'm simply overlooking something. Thanks.
Show that $\displaystyle \{x \in D_n | x^2=e\}$ is not a subgroup of $\displaystyle D_n (n\geq 3) $.
The composition of two reflections is a rotation. When you perform one reflection, the only way to get back to the original orientation via reflection is to do the same reflection again. Thus, since you have at least two (actually 3) reflections in your set, choosing any two of them will result in a rotation that is not the identity, and thus not in the set.
Edit: Sorry forgot about the 180 degree rotation again... I'm rethinking
Edit 2: Ah we can just use that $\displaystyle S_iS_j=R_{i-j}$
http://en.wikipedia.org/wiki/Dihedra...roup_structure