There are essentially two social problems addressed by Shaw in Pygmalion. The first is the perennial problem of the enormous gap between rich and poor, not just in terms of wealth and opportunity, but also in relation to culture. Henry Higgins and Eliza Dolittle may inhabit the same city and the same country, but they might as well be living on different planets, such is the huge cultural and intellectual gulf that separates them.
Even so, Higgins's experiment of turning Eliza from a humble Cockney flower-seller to a lady of quality appears to suggest that this gulf can nonetheless be overcome. The suggestion is that if the lower-classes were properly educated, then they too could talk and behave just like their alleged social betters.
Of course, such education would not necessarily take the precise form that Higgins gives it, but it would involve the upper-classes devoting themselves, through the payment of higher taxes, towards providing for a comprehensive system of state education designed to give the Eliza Dolittles of this world the chance to improve their lot in society.
The second problem dealt with in the play is that of female exploitation. There can be no doubt that the relationship between Higgins and Eliza is fundamentally exploitative, with Higgins looking upon Eliza as nothing more than an object, a guinea-pig in his latest experiment.
What we see here is a microcosm of how women in Edwardian society as a whole are treated, even those ladies of quality whose posh voices Eliza is trained to emulate. Indeed, even when Eliza has finally learned how to talk like a duchess, she still finds herself under the control of a man, an indication perhaps that a radical form of female emancipation is required in society.