Example of an infinite non-normal subgroup $H$ of non-abelian (topological)group $G$.

Question

I am trying to find Example of an infinite non-normal subgroup $H$ of non-abelian (topological)group $G$.

For instance, The subset $\mathrm{T}(n,\mathbb{R})\subset \mathrm{GL}(n, \mathbb{R})$ of invertible upper triangular matrices is a non-normal subgroup of (topological)group $(\mathrm{GL}(n,\mathbb{R}) ,\cdot)$.

Is there exists another example in group theory or topological groups?

Thanks.

Answer 1

Let $G$ be the group of symmetries of a circle. Let $H$ be the subgroup generated by the flips in "rational axes", that is, the flips in axes that make a rational angle (measured in degrees) with some reference radius. If $r$ is an irrational rotation, then $r^{-1}Hr\ne H$.

Answer 2

Sure. Consider the subgroup $H$ of matrices of the form \begin{bmatrix} 1 & a \\ 0 & 1 \end{bmatrix} inside $GL_2(\mathbb R)$. These are "horizontal" shears; if $r$ is almost any nontrivial rotation, then $r^{-1} H r \ne H$.