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Let $I$ be the group of orientation preserving symmetries of a regular icosahedron. This is a $60$ element subgroup of $SO(3)$, isomorphic with the alternating group $A_5$. It is also perfect and 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.

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1 Answer 1

up vote 5 down vote accepted

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.

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It's obvious when you put it like that (and that the quarter turn can be replaced with anything between $0$ and $\pi$, and the same can be done for the $3$- and $5$-fold symmetry). Now I wonder if the groups appearing as intersections of conjugates are necessarily cyclic? Thanks for your speedy and accurate reply! –  Mark Grant Mar 22 '11 at 6:59
    
Actually in my example with the quarter rotation both icosahedra are preserved by some more half turns around axes that join midpoints of two antipodal edges. The two icosahedra have three such axes of rotation in common. They are perpendicular. That gives a Klein four group. They have no more axes in common. So the intersection is not cyclic and not everything. It is a Klein four group. –  Wilberd van der Kallen Mar 22 '11 at 17:31

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