Thread: Cosets/normal subgroup

Can someone please explain to me normal subgroups.

if you have H a subgroup of an abelian group G and b is an element of G, then the right coset to which b belongs is same as the left coset to which b belongs. (proving Hb=bH).

also, is a subgroup of an abelian group, necessarily abelian?


Thank you

Hello

Originally Posted by hercules

Can someone please explain to me normal subgroups.

Take a group, G. then N is called a normal subgroup of G iff and ,

You can also say N is a normal subgroup of G if the sets of left and right cosets of N in G coincide, or ,

also, is a subgroup of an abelian group, necessarily abelian?

Say H is a subgroup of an abelian group, G.

Take

Then since H is a subgroup,

Since G is abelian, you can say

So by closure

So yes, a subgroup of an abelian group is abelian.

Originally Posted by hercules

if you have H a subgroup of an abelian group G and b is an element of G, then the right coset to which b belongs is same as the left coset to which b belongs. (proving Hb=bH).

There are different ways to define normality, the way I perfer (and most algebra books use) is a subgroup of such that for all and . Now suppose that is a subgroup that has this poperty. We will prove that for any , by showing . Say then but for some . And so . And the other way around is similar.