each group tries/try

It depends. You would see both forms in informal speech, or even in writing, but since you ask.......neither is correct. I would prefer

Each group tries its best to outdo the others.

Thanks, McTag

Each group tries its best to outdo the others.

Can I say Each group try their best outdo the others.

The second sentence is the better sounding but Each group try their best outdo the others is wrong as well. Each group tries its best to outdo the others, will sound better.

Most times you see the word "each," it is signaling for a singular verb form.

true but with out the word "to" included, the sentence isnt correct. Your right though, I got stuck on the missing word but actually that sentence doesnt sound bad once its corrected.

in my less than humble, most of your questions are about stilted forms of expression, try

" the groups compete"

Since the subject is "group", it is singular and so takes a singular verb, strictly speaking.

But you often see singular subjects such a "the police" or "the team" used as if they were plural- and you can see the sense in that, from a logical point of view, if not from a strict grammatical one.