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

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- Apr 28th 2010, 01:31 AMalexandrabel90subgroups
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 - Apr 28th 2010, 02:06 AMFailure
You are right, and that would be the most natural way of defining H to be a subgroup of G. But requiring for all $\displaystyle a,b\in H$ that $\displaystyle a^{-1}b\in H$ 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 $\displaystyle b=a$ you get that $\displaystyle e=a^{-1}a\in H$; thus H contains the neutral element e. And once you have shown that you can set $\displaystyle b=e$ and get that H also contains the inverse element of every one of its elements: $\displaystyle a^{-1}=a^{-1}e\in H$.