This is the 16147th most frequent Russian word.
разрешат
"Разрешат" means "they will allow" or "they will permit."
Here, "разрешат" is used in the context of granting permission in the future tense.
Учителя разрешат ученикам пользоваться учебниками на экзамене.
Teachers will allow students to use textbooks during the exam.
In this sentence, "разрешат" refers to the possibility of receiving permission from an authority or administration.
Друзья надеются, что им разрешат устроить пикник в парке.
Friends hope that they will be allowed to hold a picnic in the park.
"Разрешат" here is employed to indicate the official authorization of an event by governing entities.