Egoism
?noun 1. the habit of valuing everything only in reference to one's personal interest; selfishness ( opposed to
altruism).
2. egotism or conceit.
3. Ethics . the view that morality ultimately rests on self-interest.
Arrogance
?noun offensive display of superiority or self-importance; overbearing pride.
http://dictionary.reference.com
So a bit of yes and no. It depends on the context of egoism. If someone is egotistic, they're selfish. If they're extremely/overly selfish they are arrogant.
I've met many of both. One friend I have is very egotistic. He's shallow and thinks he's "the shit"....but he's not overbearing, it's just a trait of who he is and it's not hurting anyone.
I know another friend who IS arrogant. He thinks he's superior and more important than everyone else. He has no shame in displaying it and will do so at any opportunity, constantly picking on others flaws even when his own are pretty clear. When his own are pointed out he has some excuse why he's allowed to be imperfect, or he says that he is perfect and uses redirection to make it seem like the other person is flawed just in pointing it out.
Egoism and arrogance are the same thing but they refer to different degrees of the same thing. Egoism is the lesser, arrogance is the extreme.