I see alot of people praying in the mosque in congregation, they always go into rukoo' at the same time of the imam when he is about to say Allaahu akbar, but then, they go into rukoo' when he is about to say the first part Allaa- or do whatever he is about to do, and i feel like this is wrong, even though he they should go after he finishes saying it and goes first is it allowed to do this?
According to this and other similar hadith:
Imams has to be followed:
Narrated Anas bin Malik:
Allah's Messenger (ﷺ) (p.b.u.h) fell down from a horse and his right side was either injured or scratched, so we went to inquire about his health. The time for the prayer became due and he offered the prayer while sitting and we prayed while standing. He said, "The Imam is to be followed; so if he says Takbir, you should also say Takbir, and if he bows you should also bow; and when he lifts his head you should also do the same and if he says: Sami`a l-lahu liman hamidah (Allah hears whoever sends his praises to Him) you should say: Rabbana walakal-Hamd (O our Lord! All the praises are for You.") Sahih Al-Bukhari USC-MSA web (English) ref. Vol. 2, Book 20, Hadith 215
The word follow means going after not going besides he or she.