Let G be a finite group, $|$G$|<\backslash$infty $,$ and let LG be the vector space of complex functions f$($g$)$

on G$,$ f : G $->$ C$.$ With respect to the inner product

$$f$1|$f$2=$

$1$

$|$G$|$

X

g in G

f$1($g$)$f$2($g$),$f$1,$ f$2$ in LG $,$

LG is a finite$-$dimensional Hilbert space.

$($a$)$ Given a group element h in G$,$ associate with it a linear operator TL : LG $->$ LG defined by

TL : f$($g$)->$

h

TL$($h$)$f

i

$($g$)=$ f$($h

$1$

g