Theory and applications of finite groups, by G.A. Miller, H. F. Blichfeldt [and] L. E. Dickson.

~ 58] GROUP OF THE REGULAR ICOSAHEDRON

151

As slS22 is the transform of the inverse of siS2a, the given statement is proved. The most general product that can be formed
with the operators Si, S2 is of the form
S2aIS1S2 a2S1S2a3S... S2aX.
If the number of factors in this product exceeds six it can evidently be reduced by means of one or more of the following
equations:
SlS2Sl = S24SlS24, Ss241S = S2SlS2, (SS23)5 = 1,
S1S22S1 = S24SlS23S1S24, S23S1 = SSS2S 2S1S2.
From these equations and from the fact that
S1S22SlS23S1S2a S2-aSlS22SlS23S1
it results that all the distinct operators of (si, s2) can be written
in one of the following forms:
S2m, 52mSlS2n, S2mSlS22SlS2n, s2mSlS22S1S23s (m, n=1, 2, 5).
Hence (sl, S2) is of order 60 if we assume that sl, S2 fulfil the
equations given above.
The three sets of equations
12 =S2,5 (SS2)3 -1, S12 =S23 =(S1S2)5- 1, S13=S25=(S1S2)2=1
are evidently equivalent and hence each of them defines the
icosahedral group. In fact, if we should assume that the operators sl, S2 merely satisfy any one of these sets of equations, they
would generate the icosahedron group unless both of them
were the identity. Hence we have the theorem:
If the three numbers 2, 3, 5 are the orders of two operators and
of their product, these operators must generate the icosahedron
group irrespective of which of these three numbers is the order
of the product.
EXERCISES
1. If two operators merely satisfy the equations defining the tetrahedral group they may generate the cyclic group of order 3, and if they
merely satisfy the equations defining the octahedral group they may
generate the symmetric group of order 6 or the group of order 2.
2. If a group of order 12 contains no invariant operator of order 2 it
must be tetrahedral.