Problems in Wikipedia about symmetry group of the non-equilateral isosceles triangle.

Question

I think this wikipedia article - https://en.wikipedia.org/wiki/Symmetry_group#Two_dimensions - is wrong when it states that "$D_2$, which is isomorphic to the Klein four-group, is the symmetry group of a non-equilateral isosceles triangle (...)", since the symmetry group of the non-equilateral isosceles triangle would be $D_1$, like the symmetry group of the letter ${\bf A}$ (order 2).

Also, this article - https://en.wikipedia.org/wiki/Isosceles_triangle - says that the symmetry group of the isoceles triangle is $D_2$ ($Dih_2$), and that this group has order $2$. But the order of a dihedral group $D_n$, as said in the previous article, is $2n$, so $D_2$ must have order 4, right?

Answer 1

Yes, the symmetry group of a non-equilateral isosceles triangle has order 2. And, depending on convention, $D_n$ either has order $n$ or $2n$.

Answer 2

As explained on the Wikipedia page on dihedral groups, there are two different conventions in which the group denoted by $D_n$ can be either

• the dihedral group of order $n$, or
• the automorphism group of a regular polyhedron with $n$ sides (which has order $2n$).

Update: The above is addressing only the last (and only) question in your post's body. You are correct in pointing out that the symmetry group of a (non-equilateral) isosceles triangle isn't the Klein-four group, though.

Update 2: The Wikipedia article on symmetry groups has been fixed since you asked this question. An example of plane figure with the Klein-four as symmetry group is a non-equilateral rectangle. The four symmetries are: the identity, the reflections that swap the two pairs of opposite sides, and the rotation by $\pi$.