Is it permissible for a sister to travel alone to Australia for study for a period of 3 years without a mahram, knowing that a trustworthy relative (her other sister's husband) is there?

Scholars have differed regarding the ruling on a woman traveling for obligatory Hajj and Umrah with a safe company but without a mahram. However, the majority are of the opinion that it is not permissible for her to travel for anything other than obligatory Hajj and Umrah without a mahram. Therefore, it is not permissible for your sister to travel abroad for study without a mahram, as the condition of a mahram is specific to travel. As for residence, the condition is safety from fitna (temptation/strife), and residence in lands of disbelief is not permissible except under specific conditions and regulations.