Normal subgroup created by a bunch of elements

Question

if I have a finite group $G$ and a bunch of elements that are the elements of a set $A$. How can I systematically calculate the smallest normal subgroup of $G$ that contains $A$? I am rather interested in a practical way how this can be done?

Answer 1

If you are given a group table (or something other way of working concretely with elements) and a starting subset, one way might be to use the following algorithm:

Let $S_0$ be the given set of elements, their inverses, and the identity. Let $i=0$.

(i) Let $S_i'$ be the set of all conjugates of elements in $S_i$.

(ii) Let $S_{i+1}$ be the set of all possible products of pairs of elements in $S_i'$.

(iii) If $|S_{i+1}| = |S_i|$, stop. Otherwise, increment $i$ by $1$ and return to (i).

We need to verify that this procedure builds a normal subgroup. We first note the following:

(0) $e \in S_0$

(1) $a, b \in S_m \implies ab \in S_{m+1}$.

(2) $a\in S_m \implies a^{-1} \in S_m$.

(3) $a \in S_m$, $g \in G \implies g a g^{-1} \in S_{m+1}$.

Of the above, (2) is the only one that is perhaps non-trivial. This can be shown by induction. Now, note the algorithm does terminate, since the group is finite. Suppose that the terminal set is $S_{k+1}$. Then $S_{k+1} = S_{k}$, and (0)-(3) above show $S_{k}$ is a normal subgroup.

Answer 2

Try the intersection of all normal subgroups that contains your subset $A$.