(Related [question](https://mathoverflow.net/questions/58320/intersections-of-conjugates-of-lie-subgroups))


Let $I$ be the group of orientation preserving symmetries of a regular [icosahedron](http://en.wikipedia.org/wiki/Icosahedral_symmetry). This is a $60$ element subgroup of $SO(3)$, isomorphic with the alternating group $A\_5$. It is also [perfect](http://en.wikipedia.org/wiki/Perfect_group) and [self-normalizing](http://en.wikipedia.org/wiki/Self-normalizing) in $SO(3)$.


For each $g\in SO(3)$ the conjugate ${}^gI=gIg^{-1}$ is the group of rotations which leave invariant a rotated icosahedron. My question concerns which groups can appear as intersections of conjugates. In particular, can anyone supply a proof or disproof of the following statement?



> 
> For any $g\in SO(3)$ the group ${}^gI\cap I$ is either trivial, or equal to $I$.
> 
> 
> 


This elementary group theory question arose when studying numerical homotopy invariants of the Poincaré sphere $X=SO(3)/I$. In particular, I would like the fixed point sets of the two-point stabilisers of the standard action of $SO(3)$ on $X$ to be path-connected.


Thanks in advance.



Intersections of conjugates of the icosahedral group in SO(3) 
Rotate a quarter turn around the axis passing through the midpoints of two antipodal edges.
That gives a different copy of the original icosahedron. A half turn preserves both
icosahedra. So the statement is wrong.


