Understanding the concept

Question

I have seen the statement-"Kleins four group is isomorphic to the dihedral group of order 4". I am not getting how to get the dihedral of order 4 as for the dihedral having order four it should be regular 2-gon. What are the elements of this dihedral?

Answer 1

For the purpose of defining dihedral groups, there is no loss, if we think of regular $2$-gon as follows: consider a circle, and fix two opposite end-points $P,Q$. Then there are two paths from $P$ to $Q$, both of same length, and we can call it regular $2$-gon.

Then this regular $2$-gon has four symmetries: let the points $P,Q$ be like south-pole and north-pole. Then vertical reflection, horizontal reflection, $180^o$ rotation around center, together with identity will give the Klein-4 group.