You haven't specified a chapter, but Tom's actions in chapter 1 do help to reveal these aspects of his character. Nick describes Tom's behavior when Tom shows Nick his house. He says,

Turning me around by one arm, [Tom] moved a broad flat hand along the front vista . . .. "It belonged to Demaine, the oil man." He turned me around again, politely and abruptly. "We'll go inside."

Tom's actions begin to paint his character as domineering: he puts his hands on Nick and physically turns Nick where he wants Nick to look. Twice. It's as though he is so controlling, so used to others submitting without question to him, that he feels entitled to manipulate and manhandle people. He doesn't ask Nick if he wants to go inside; he tells him that's what they're doing. In his arrogance, Tom clearly feels entitled to speak this way to others. Later, Tom does something similar. Nick says,

wedging his tense arm imperatively under mine, Tom Buchanan compelled me from the room as though he were moving a checker to another square.

Tom steers Nick from room to room, and Nick feels like an object, a toy, in the hands of this domineering and arrogant man.

Further, though this man could hardly have any reason to feel badly about society—in which he lives quite richly and without day-to-day cares—Tom says that "'Civilization's going to pieces,'" and he proceeds to explain that it's because of the advancement of people of color. He fears that "'the white race will be . . . utterly submerged,'" that it will lose power and control. Again, this shows the extent to which he must dominate. He is a blatant racist, one who must see other races as lesser and deserving of domination because he can only conceive of himself, of his race, as superior.

Tom Buchanan's basic character is certainly domineering and arrogant. However, you have not specified which chapter in The Great Gatsby you are querying about.