TITLE: Finitely presented group with fewer relations than generators.
QUESTION [8 upvotes]: I some how feel that, any finitely presented group with less relations than generators has to be an infinite group.
In one of my questions, in an answer I have seen a group which is finitely presented but it is not finite.
$G=\langle a,b\mid a^2=b^2=1\rangle$ this group is infinite. I see that if there is at least one more relation in between $a$ & $b$ then the group is finite.

I claim that if there are fewer relations than generators of a finitely presented group, then the group has to be infinite.

I would like to see is it true in general??
Is any finitely presented group with fewer relations than generators an infinite Group?
Please suggest some idea to see if this is true/false?

REPLY [3 votes]: I think you are somehow looking for Deficiency of a group and here. I had a course before about the presentation of groups and in my notes it's stated a Theorem as follows. I couldn't find it through web. Sorry.

Theorem (Mc Donald): Every group $G$ with positive deficiency is infinite.

You may consult here for more examples. Of course, the converse of the above theorem is not right in general. See here and here for finding the counter-examples.