Does divorce occur if the brother uses the Internet, and the swearer swore an oath of divorce upon his brother not to use it? And what is the expiation for this oath?

If your brother uses the internet, your wife is divorced, whether you intended to cause the divorce or not. This is the view of the majority of scholars. If the divorce that occurred is less than three pronouncements, you may take your wife back during her iddah (waiting period).

Some scholars, like Sheikh al-Islam Ibn Taymiyyah, hold the view that if swearing by divorce is not intended to be a conditional divorce, but rather a threat or an emphasis, then its ruling is like swearing by Allah. If the swearer breaks his oath, he is obligated to pay an expiation for an oath, and the divorce does not occur.

What matters in breaking the oath is your intention with what you uttered; if you intended to prevent him from using the network at a specific time or in a specific manner, then you do not break your oath if he uses it at other times, for intention in an oath specifies the general and restricts the absolute.

It is recommended to present the issue to scholars known for their jurisprudence and religiosity in your country. And know that the permissible oath is swearing by Allah Almighty, whereas swearing by divorce is among the oaths of the dissolute, and one should be cautious of it.