7・19 Given 3 lines in space, where any two lines are neither parallel nor coincident, how many axes of symmetry can this figure have at most?


[Sol] There are at most 9 axes of symmetry.
Consider the 3 axes of the Cartesian coordinate system in space. In this case, each axis itself is 1 axis of symmetry. In addition, the two angle bisectors of each pair of axes are also axes of symmetry, hence there are a total of 9 axes of symmetry.

On the other hand, let the 3 given lines be $l_{1}, l_{2}, l_{3}$. Then the axes of symmetry of the figure can be divided into the following two categories:
(1) Two lines are symmetric to each other about the axis of symmetry, while the third line is symmetric to itself about the axis of symmetry. Since there are only two axes of symmetry for two lines in space that are neither parallel nor coincident, there are at most 6 axes of symmetry in this category.
(2) All three lines are symmetric to themselves about the axis of symmetry. Since a line is symmetric to itself about an axis of symmetry if and only if it is perpendicular or coincident with the axis of symmetry, and any two non-parallel lines have exactly 1 common perpendicular, there are at most 3 axes of symmetry in this category. At this time, each line is the common perpendicular of the other two lines.

If the 3 lines have 1 common perpendicular, then these 3 lines cannot have any other axes of symmetry.
In summary, this spatial figure has at most 9 axes of symmetry.