Mathematical_Recreations-Kraitchik-2e

Mathematical Recreations
which has had a profound influence on all branches of mathematics.
It will be worth our while to see what it can contribute
to our present problem.
Following is the multiplication table of the group - a
table which gives in each row and column the product of the
element at the left of the row by the element at the head of
the column.
I R R2 R3 S SR SR2 SR3
R R2 R3 I SR3 S SR SR2
R2 R3 I R SR2 SR3 S SR
R3 I R R2 SR SR2 SR3 S
S SR SR2 SR3 I R R2 R3
SR SR2 SR3 S R3 I R R2
SR2 SR3 S SR R2 R3 I R
SR3 S SR SR2 R R2 R3 I
From it we can see that the group contains other groupscalled
subgroups. There are in fact just these 10 subgroups,
including the group itself as a subgroup:
(1) The identity: I.
(2) Rotations through 2 right angles: I, R2.
(3) Reflections about the middle vertical line: I, S.
(4) Reflections about the middle horizontal line: I, SR2.
(5) Reflections about the downward diagonal: I, SR.
(6) Reflections about the upward diagonal: I, SR3.
(7) Rotations through one right angle: I, R, R2, R3.
(8) Reflections about either middle orthogonal line: I, R2, S,
SR2.
(9) Reflections about either main diagonal: I, SR, SR3, R2.
(10) The whole group.
Every solution of the problem of the rooks has its corresponding
group - the largest subgroup of the given group
which leaves it invariant (that is, unchanged). In this way
we can classify the various solutions. It will be found that