The simplest combinations are those characterized by the presence of only one axis which may be a proper axis or an inversion axis, or simultaneously proper and improper axis.

One proper symmetry axis point group:

One inversion axis point group:

One proper and improper axis point group:

/, /, /, /, /, /, ...

Let us see the arrangement of the set of symmetrically equivalent points for the various cases:

/ : is a trivial axis, thus /.

/ /m where m is perpendicular to the axis.

/ , since = contains .

/ /m where m is perpendicular to the axis.

/

//m where m is perpendicular to the axis.

If two rotation symmetry axes L1 and L2 coexist and intersect in O, the first axis making the object in P symmetrically equivalent to the object in Q, the second axis making the object in Q equivalent to the object in R, a third axis L3 must exist making the object in R equivalent to the object in P.
If L1 is a proper axis while L2 is an inversion one, then L3 will be an inversion axis.

The only allowed combinations are

n22 , 233, 432, 532

where n may assume any integral value. The angles between the axes are as follows:

222	90(2 2)	90(2 2)	90(2 2)
322	90(2 3)	90(2 3)	60(2 2)
422	90(2 4)	90(2 4)	45(2 2)
622	90(2 6)	90(2 6)	30(2 2)
233	54 44'08"(2 3)	54 44'08"(2 3)	70 31'44"(3 3)
432	35 15'52"(2 3)	45(2 4)	54 44'08"(4 3)

New point groups may be found by allowing symmetry axes to be inversion axes.