We think it's a good idea to allow teenagers to study in groups?

allow前的to是短语“it’s.....to do sth” 意思是“做什么是…的”
study后面的to是“allow to do sth” “允许做某事”

to 后表示目的，说明是什么样的好主意。

好好想想先。。。

"to" is a preposition. This is part of the English language Grammar.
you need to use "to" and a verb, combine to produce an adjective, to describe the noun "idea"

study前的to是“allow ... to ...” 的词组允许做某事。

allow前的to是“it's a good idea to ...”类似于指示代词，表明目的。。。