Determining if Subset is a Subgroup by using Group Presentation

[Bacle]

Hi, Algebraists:

Say I'm given a group's presentation G=<X|R>, with

X a finite set of generators, R the set of relations. A couple of questions, please:

i)If S is a subset of G what condition must the generators of

S satisfy for S to be a subgroup of G ? I know there is a condition

that if for any a,b in S, then S is a subgroup of G if ab^-1 is in S, but

I am tryng to work only with the generating set.

ii) If A,B are known to be subgroups of G; G as above: what

condition do I need on the generators of A,B respectively,

in order to tell if A is a subgroup of G? Is inclusion enough?

Thanks.

 

[lugita15]

I think a subset of a group which is generated by a set of generators is automatically a subgroup. I may be wrong though.

 

[Bacle]

I got this one: a necessary and sufficient condition is that every generator of A can be
written as a word in B.