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Suppose that T and T are alternative coset representatives for ST1 and 8T2 respectively, so that T E ST1 and T c ST2. It has to be proved that S(TT) = S(T1T2). As T E ST1 and T C ST2, there exist S,S' E S such that T = ST1 and T = S'T. Then TT = STIS'T2. But T1S r C TIS, so, as S is an invariant subgroup, T1S r E ST1. Consequently there exists an S' e S such that T I S ' = S'T1.
Then TT = (SS')(TIT2), so that TT E S(TIT2) and hence, by property (f) of the first theorem of Section 4, $(TT)= S(TIT2). SCay = 8C2d = 8C2c). 6 Homomorphic and isomorphic mappings Let G and G be two groups. A 'mapping' r of G onto is simply a rule by which each element T of G is assigned to some element T = r of G', with every element of Gt being the 'image' of at least one element of G. 7) for all T1, T2 E , then r is said to be a 'homomorphic' mapping. 7) the product of T1 with T2 is evaluated using the group multiplication operation for G, whereas on the left-hand side the product of r with r is obtained from the group multiplication operation for Gp. Dyn az (Yl,., Yn ) 1 n and dT - dyn ar (Yl, 999 Yn) dyl.
1 n are finite., Yn) and a t ( y 1,.,y) are equal, so that the integrals are both left- and rightinvariant, then G is said to be 'unimodular', and one may write dIT = dT = dT and (71(Yl,.,yn). 14) can be generalized in the obvious way to include a sum over the components. The significance of the distinction between compact and non-compact Lie groups lies in the first two of the following theorems, the first of which was originally proved by Peter and Weyl (1927).