Thread: subgroups

what does it mean by to be a subgroup H of G where G is a group, then for all a,b elements of H, a^-1 b is an element of H.

i thought that to be a subgroup, elements of H have to be in G and it has to obey the conditions of a group?

thanks

Originally Posted by alexandrabel90

what does it mean by to be a subgroup H of G where G is a group, then for all a,b elements of H, a^-1 b is an element of H.

i thought that to be a subgroup, elements of H have to be in G and it has to obey the conditions of a group?

You are right, and that would be the most natural way of defining H to be a subgroup of G. But requiring for all that is just an equivalent, and apparently some think more useful way of defining the same concept (more useful in the sense that proving a subset H of G to be a subgroup requires proving only that single statement).

Assuming that funny definition of being a subgroup, if you set you get that ; thus H contains the neutral element e. And once you have shown that you can set and get that H also contains the inverse element of every one of its elements: .