By J. S. Lomont (Auth.)

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Another group of interest is the icosahedral group «X, which is a molecular point group. It is the simple insolvable group of order 60, and has ñve classes. Also, y is a perfect group. Its generators and defining relations are Αλ* = A22 = A32 = I (Α^ύ* = (A2AJ* = I (AXA^ = L The groups C 2 , Dn, and 0 are simply reducible, whereas CH (n ^ 2), T, and J* are not. It is easily seen that the set of all 3-dimensional rotation matrices is a group (infinite) if the group operation is taken to be matrix multiplication.

1) Order (Gx X G2) = g±g2. (2) Gx x G2 is isomorphic to G2 x Gv (3) Both Gx and G2 are normal subgroups of Gx x G2. (4) The direct product of two abelian groups is abelian. (5) The direct product of two solvable groups is solvable. (6) The number of classes of Gx x G2 is rx r2. (7) If Cx and C2 are classes of G± and G2, respectively, then the set of elements Cx x C2 is a class of Gx x G2, and every class of Gx x G2 is of this type. (8) If Hx and H2 are normal subgroups of G1 and G2, respectively, then H1 x H2 is a normal subgroup of Gx x G2, and (G1 X G2)¡{H1 x H2) is isomorphic to {GX¡H^ x (G2\H2).

This simple transcription of formulas from matrix groups to representations is possible because in a homomorphism the number of elements mapped on an element of the homomorph is the same for all elements of the homomorph. We might at this point state a theorem due to Ito on the dimension of any irreducible representation of a group, which is somewhat sharper than the one we stated in connection with matrix groups. Theorem. The dimension d of any irreducible representation of a group G must be a divisor of the index of each of the maximal normal abelian subgroups of G (where by a maximal normal abelian subgroup is meant a normal abelian subgroup which is not a proper subgroup of any other normal abelian subgroup).