"Lev Nicolaevich!" cried Parfyon, before he had reached the next landing. "Have you got that cross you bought from the soldier with you?"
"Yes, I have," and the prince stopped again. […] "Give it to me," said Parfyon.
"Why? do you—"
The prince would rather have kept this particular cross.
"I'll wear it; and you shall have mine. I'll take it off at once."
"You wish to exchange crosses? Very well, Parfyon, if that's the case, I'm glad enough—that makes us brothers, you know."
The prince took off his tin cross, Parfyon his gold one, and the exchange was made.
Parfyon was silent. With sad surprise the prince observed that the look of distrust, the bitter, ironical smile, had still not altogether left his newly-adopted brother's face. At moments, at all events, it showed itself but too plainly. (2.4.28-36)
Why does Rogozhin want to be cross-brothers with Myshkin? Why do they hang out together as much as they do? Do you think, in a different situation without a woman to fight over, they would've been friends?