<div class="center"> # Multiplicative Property of Characters Let $ G $ be an arbitrary group. Let $ a \in G $ and $ b \in G $. $$ f(ab) = f(a) f(b). $$ Let $ \cdot $ denote the group operator and $ \times $ denote the multiplication operator for complex numbers. $$ f(a \cdot b) = f(a) \times f(b). $$ <div style="text-align: left; display: inline-block"> Note: * The variables $ a $ and $ b $ represent elements of a group $ G $. * The character $ f $ is a complex-valued function. * In the notation $ f(a) $, $ a \in G $ and $ f(a) \in \mathbb{C} $. * If we represent the group operator as $ \cdot $, then $ f(ab) = f(a \cdot b) $. * The notation $ f(a) f(b) $ represents multiplication of two complex numbers $ f(a) $ and $ f(b) $. </div> </div>