The lord is introduced praising his (well-trained) hunting dogs. In retrospect, of course a person like that would enjoy having plays in which women are tamed performed.
When Katharina hits Petruchio in the fiery and hilarious dialogue of their first meeting, he warns her that if she does that again, hell retaliate. Kate answers with a remark about how if he beats her he will be no gentleman. Now that was really bold for Kate: beating Hortensio, who after all is (or pretends to be) a servant while shes the lady of the house is a thing of little consequence, but striking the suitor whos equal to her in status and approved by her father in a very patriarchal society is quite different. Petruchio would be in his right hitting her back and her dad would only say serves you well. Considering how Petruchio, despite his bombastic language and threats, never resources to physical aggression, Kates instinct seems right from start: hes not the kind who would hit a girl. It adds to the brilliance that she hits him explicitly saying that she will try if hes a gentleman as he claims: he passes the test.
Theres a lot of debate over if this play is indeed an affirmation of sexist ideas or a subversion of them, or something in-between. The critical point seems to be if Katharinas famous last speech is intended to be serious or if its an ironic internal joke between her and Petruchio. The fact that Petruchio and Katharina are having the time of their lives in the last scene, shocking everybody with her transformation and the foolish duty demonstration, and Katharinas discourse on the feminine submission is so exaggerated and excessive, seems to support the latter interpretation.
Willy's good old Ode to Wife Beating? I think that goes in several places, but it's very apparent even slightly into the play.
Or Ode Against Wife Beating, given how Petruchio emphasizes how he can tame his wife without beating her. Still not compatible with modern values, but...
Though the concept of marital rape was unheard of in the 16th century, its evident that Petruchio does not want to have sex with Kate before she warms up to him and consents. Everybody remembers the charming kiss me Kate scene, when Petruchio tries to manipulate Kate into giving him a kiss, but in an inversion, its she who learns how to manipulate her childish and hyperbolic husband both see through the others actions and delight in them. It foreshadows the full sexual consent later. Almost the last words he says in the play are Come, Kate, well to bed, i. e., they will have their nuptials at last!