What Is the Smallest Normal Subgroup of a Group Containing a Given Subset?

[norajill]

1) Let X be anon empty subset of a group G .prove that there is a smallest normal subgroup of G containing X
ii)what do we call the smallest normal subgroup of G containing X

 

[Dick]

If you want to know what something is called, I think you should look it up. If you want to know how to prove something, you should show us at least an attempt.

 

[Architeuthis Dux]

1) To prove that there is a smallest normal subgroup of G containing X, we first need to understand the properties of normal subgroups. A normal subgroup is a subgroup that is invariant under conjugation, meaning that for any element x in the subgroup and any element g in the group G, the conjugate of x by g (i.e. g^-1xg) is also in the subgroup. This means that normal subgroups are closed under conjugation and are necessary for the concept of quotient groups.

Now, let X be a non-empty subset of a group G. We can define the set of all normal subgroups of G that contain X as N = {N | N is a normal subgroup of G and X ⊆ N}. Since G itself is a normal subgroup of G and X is a subset of G, N is non-empty. By the well-ordering principle, N has a smallest element, denoted by N*. We claim that N* is the smallest normal subgroup of G containing X.

To prove this, we need to show that N* satisfies two properties: (i) it is a normal subgroup of G, and (ii) it is the smallest normal subgroup of G containing X.

(i) To show that N* is a normal subgroup of G, we need to show that for any element g in G, the conjugate of N* by g (i.e. g^-1N*g) is also in N*. Since N* is the smallest element of N, it contains all normal subgroups of G that contain X, including g^-1N*g. Therefore, g^-1N*g is also in N*, making N* closed under conjugation and thus a normal subgroup of G.

(ii) To show that N* is the smallest normal subgroup of G containing X, we need to show that for any normal subgroup N of G that contains X, N* is a subgroup of N. Since N is a normal subgroup, it is closed under conjugation and thus contains all conjugates of its elements. This means that N* is also a subgroup of N, as it contains all elements of X and their conjugates. Therefore, N* is the smallest normal subgroup of G containing X.

ii) The smallest normal subgroup of G containing X is called the normal closure of X and is denoted as <X>. It is also sometimes referred to as the subgroup generated by X.