where N is a cyclic group whose order is a factor of d and G' is a subgroup of PGL(2, C), i.e., a cyclic group [Z.sub.m], a dihedral group D2m, the

tetrahedral group [A.sub.4], the octahedral group S4 or the icosahedral group [A.sub.5].

For example, it was shown in [20] that the alternating group [A.sub.4] and the binary tetrahedral group B [congruent to] [SL.sub.2](GF(3)) have isomorphic endomorphism monoids.

* the alternating group A4 (also called the tetrahedral group);

* the binary tetrahedral group B = {a, b | [b.sup.3] = 1, aba = bab) (|B| = 24);

The alternating group [A.sub.4] (also called the tetrahedral group) and the binary tetrahedral group B = (a,b|[b.sup.3] = 1, aba = bab) are the only groups of order less than 36 that are not determined by their endomorphism monoids in the class of all groups [9,20-23] (for some groups of order 32 the proofs are under publishing).

The 24 quaternions of the binary tetrahedral group [3, 3, 2] are contained already in the above 120 icosians.

For example, the electron family has the discrete symmetry of the binary tetrahedral group and the electron is one of its two possible orthogonal basis states.

The alternating group [A.sub.4] (also called the tetrahedral group) and the binary tetrahedral group B = <a, b | [b.sup.3] = 1, aba = bab> are the only groups of order less than 32 that are not determined by their endomorphism semigroups in the class of all groups [12].

where [C.sub.3] is the cyclic group of order 3 and A4 is the alternating group of order 12 (the tetrahedral group).

(3) the endomorphism semigroups of G and [G.sub.13] are isomorphic if and only if G = [G.sub.13] or G = [C.sub.3] x B, where B = <a, b | [b.sup.3] = 1 , aba = bab> is the binary tetrahedral group;

It was proved in [12] that among the finite groups of order less than 32 only the alternating group A4 (also called the

tetrahedral group) and the binary

tetrahedral group <a; b | [b.sup.3] =1; aba = bab> are not determined by their endomorphism semigroups in the class of all groups.

All

tetrahedral groups are considered to be identical, which is obviously true for the perfect crystal.