Re: Group generated by rotations around two axes

Jim Heckman wrote:

>> _Problem_: If _r_ is a line in R^3 which passes through the origin and
>> if _p_ is a natural number, let R(v,p) be the group of all rotations
>> around _r_ whose angle is a multiple of 2pi/p. Given two lines _r_ and
>> _s_ and two natural numbers _p_ and _q_ greater than 1, when is it true
>> that the group generated by R(r,p) and by R(s,q) is finite? When it is
>> not finite, is it always a dense subgroup of SO(3,R)?
>>
>> For instance, when _r_ and _s_ are orthogonal, then the group generated
>> by R(r,4) and by R(s,4) is finite; it's the group of the rotations of a
>> cube.
>
> The only non-trivial finite subgroups of SO(3,R) are, up to
> conjugacy:
>
> 1) A cyclic group, around a single axis, of order n for each
> n >= 2.
>
> 2) A dihedral group of order 2n for each n >= 2. Each of these
> groups is the composition of a group C of type 1 and order n,
> with rotations of pi around each of n axes perpendicular to the
> axis of C and spaced at multiples of 2pi/n.
>
> 3) The three rotational symmetry groups of the Platonic solids: the
> tetrahedral group ~= A_4, the cubic/octahedral group ~= S_4, and
> the dodecahedral/icosahedral group ~= A_5.
>
> So G = <R(r,p),R(s,q)> is the smallest of the above that contains
> the 2 generators. If none of them do, G is infinite.

Thanks. Meanwhile, I proved that a nonabelian subgroup of SO(3,R) is
either finite or dense.

Best regards,

Jose Carlos Santos

.

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