Discussing whether or not Myrtle has "any right to mention Daisy's name," Tom breaks Myrtle's nose when she indignantly insists that she does have such a right, even going as far as to taunt him by repeating Daisy's name.

This sudden act of violence illustrates how Tom views Myrtle as nothing more than a sexual object. He has no problem using her however he wants and manipulates her using whatever means will get the job done quickly. This includes physical force as well as emotional manipulation. Earlier in the chapter, Nick learns that Tom has been lying to Myrtle about his wife being Catholic so that she won't bother him too much about leaving his current marriage right away. This gives Myrtle, who already puts more emotional stock in their relationship than Tom does, some hope that they will marry and proves effective in keeping her by his side.

That the violence is over alleged disrespect to Daisy also shows that Tom views Myrtle as inferior to his wife. It must also be remembered that Myrtle is not only just Tom's mistress; she is also from a lower class. Being a garage mechanic's wife, Myrtle is far beneath an old-money socialite like Daisy. She has no hope of being taken seriously as a marital option for an elitist like Tom, but he uses false hope to keep her in his bed and under his control.

In chapter 2, Tom and Nick head to the city together, and Tom wants Nick to meet his mistress, Myrtle. Nick spends part of the evening talking to a woman named Catherine, Myrtle's sister. She fills Nick in on Tom and Myrtle's relationship, mentioning that the two would get married if Tom could get a divorce. Catherine explains that since Tom's wife is Catholic, he is having a hard time leaving her; Nick is Daisy's cousin, so he knows that she isn't Catholic and is surprised by the "elaborateness of the lie." Tom has seemingly concocted this story to keep Myrtle around, and she believes that one day the two of them will leave their spouses and marry each other.

Time passes at the party, and for whatever reason, Daisy comes up in a conversation between Tom and Myrtle. Tom has evidently forbidden Myrtle to say Daisy's name, and Myrtle taunts him by repeating the name "Daisy" over and over. In response, Tom unexpectedly uses his open hand to break Myrtle's nose. Blood goes everywhere.

This act shows that Tom is a violent man with no respect for Myrtle. He is using her for his own pleasures, and he does not love or respect her, despite what she believes about their future together. Instead, Tom's alliance is clearly with his wife Daisy, which is why he refuses to allow his mistress to speak of her. Despite his affair, this scene proves that Tom holds Daisy in a higher regard than Myrtle, and he doesn't flinch to use violence to remind Myrtle where his loyalties lie.

Towards the end of chapter 2, Tom Buchanan and Myrtle Wilson get into a heated argument over whether she has the right to mention Daisy's name. When Myrtle begins repeating Daisy's name over and over, Tom makes a "short deft movement" and breaks her nose with his open hand. Tom's violent reaction portrays him as an aggressive, insensitive man who is impulsive and lacks discretion. Tom's display of violence also reveals his feelings towards Myrtle Wilson. In Tom's eyes, Myrtle is simply an object with which he can do whatever he wishes.

Tom is not shy about his affairs. He does not view Myrtle as a potential spouse and appears to have no plans of leaving Daisy for her. Tom shows remarkably more discretion around Daisy and would likely never raise his hand to her. While Tom's respect for Daisy is certainly questionable, he seems to treat her with more respect, possibly because of her background and social status. Given Tom's elitist personality, he could never view Myrtle as Daisy's equal. Sadly, Myrtle views Tom as a way out of a dull marriage and believes he is willing to leave Daisy. Nick and the audience realize that Tom is taking advantage of Myrtle, and Tom reveals his lack of respect for her by breaking her nose for mentioning Daisy's name.

Myrtle insists on saying Daisy's name, again and again, even after Tom has apparently told her to stop.  

"Making a short deft movement, Tom Buchanan broke her nose with his open hand."

He breaks his mistress's nose for speaking his wife's name.  This shows that Tom views Myrtle much differently from the way he views his wife.  He's elitist, and so he treats his wife, who is of much higher worth and value, much better than he treats his mistress, who is of a lower social class and is relatively poor.  Daisy is indispensable to him while Myrtle is very much replaceable.  There will always be women lining up to be with Tom because he's so rich and young and handsome.  Back in Chapter One, when Daisy insisted on repeating a word—"hulking"—that Tom doesn't like, he merely "object[s] [...] crossly" at her use of the word.  He doesn't slap her or even get angry at her.  This episode shows, quickly, how ingrained class differences are in Tom's mentality; he doesn't seem to think twice—or at all—before striking Myrtle because she is so far beneath him.

At a party in their apartment, Myrtle keeps saying the name "Daisy" over and over. Tom hits Myrtle across the face, breaking her nose. This is the second time that Tom's penchant for hurting women is very real. When Nick first meet Daisy, she is complaining that Tom is a "brute" and he hurt her little finger. However, the incident was only alluded to by Daisy. In Chapter Two, Tom's explosive anger is actually revealed as Nick and the others in the apartment see him hit Myrtle in order to stop her taunting.