Does a divorce oath take effect due to sending a Facebook join invitation to a brother and then immediately canceling it, knowing that the husband forbade his wife from her brother and swore an oath of divorce if she contacted him, and no actual contact occurred?

The fatwa we adhere to is that an oath sworn by divorce (talaq) results in divorce upon breaking the oath. This is the view of the majority of scholars. However, Shaykh al-Islam Ibn Taymiyyah holds that an oath sworn by divorce, when the intention is not to effect divorce but rather to threaten or emphasize a matter, takes the ruling of an oath sworn by Allah. Thus, if the oath is broken, the one who swore it must pay an expiation for an oath (kaffarat yameen), and no divorce occurs.

The reference for determining what constitutes breaking the oath is the intention of the one who swore it concerning what he uttered. If the husband intended to prohibit communication with the brother by any means, then breaking the oath occurs by messaging him. However, if he intended to prohibit speaking to him but not messaging him, then messaging does not constitute breaking the oath. And if the invitation did not reach your brother at all and was deleted before he became aware of it, then the oath was not broken by that.

If the invitation reached your brother, it appears that it falls under the meaning of messaging, and thus the oath is broken. However, if you sent the invitation while forgetful of your husband's oath, some scholars have held that the oath is not broken in such a case.