How is this subgroup normal?

Question

Let $G$ be a group, and let $U$ be a subset of $G$. Let $\hat{U}$ be the smallest subgroup of $G$ containing $U$. Then $\hat{U}$ is the intersection of the collection of all the subgroups of $G$ containing $U$. (Right?) This collection is obviously non-empty as $G$ itself is a subgroup of itself which contains $U$.

Now my question is this:

If $gug^{-1} \in U$ for all $g \in G$ and $u \in U$, then is the subgroup $\hat{U}$ a normal subgroup of $G$?

If the answer is yes, then one way of proving this is to find a homomorphism $ \phi \colon G \to G$ with kernel equal to $\hat{U}$.

If the answer is no, then what counter-example can we give?

Answer 1

The condition $gug^{-1} \in U$ for all $g\in G$ and $u\in U$ means that we have $gUg^{-1} = U$ for all $g\in G$. For if we had $gUg^{-1} \subsetneq U$ for some $g$, let $D = U \setminus gUg^{-1}$, then

$$g^{-1}Ug = g^{-1}\left( gUg^{-1} \cup D\right)g = g^{-1}gUg^{-1}g \cup g^{-1}Dg = U \cup g^{-1}Dg,$$

and $g^{-1}Dg\neq\varnothing$, but $U\cap g^{-1}Dg = \varnothing$, since the conjugation $x\mapsto g^{-1}xg$ is a bijection.

Thus, for any subgroup $H\subset G$, we have

$$U\subset H \iff g^{-1}Ug\subset H \iff U \subset gHg^{-1}$$

for all $g\in G$. In particular, since $U\subset \hat{U}$, we have $U \subset g\hat{U}g^{-1}$ for all $g\in G$, and since $\hat{U}$ is the smallest subgroup containing $U$,

$$\hat{U} \subset g\hat{U}g^{-1}$$

for all $g\in G$, which is another way to characterise the normality of a subgroup. Thus indeed, $\hat{U}$ is normal.

Answer 2

We may as well assume $e \in U$ because $e \in \hat{U}$. Then the closure of $U$ contains the set of all products $ab^{-1}$ for all $a,b \in U$. Then maybe this construction has to be repeated on the set of products obtained (I'm not sure), but clearly if the construction is repeated indefinitely and you take the union then you will have the subgroup $\hat{U}$. So all we have to do is show that if $a,b \in U$ then $g (ab^{-1}) g^{-1} \in \hat{U}$ for all $g \in G$. But note that $g (ab^{-1}) g^{-1} = (gag^{-1})(gbg^{-1})^{-1}$, which is a product of the form $a'b'^{-1}$ where $a',b' \in U$ and thus is contained in $\hat{U}$. So $\hat{U}$ is normal.

Answer 3

This is true. Let's call a subset of $G$ normal iff it is closed under conjugation. Let $T=U\cup\{u^{-1}|u\in U\}$. Let $g\in G$ be any element. If $t\in T$, then either $t\in U$, therefore $gtg^{-1}\in U$ or $t=u^{-1}$ for some $u\in U$, therefore $gtg^{-1}=gu^{-1}g^{-1}=(gug)^{-1}$ since $gug^{-1}\in U$, then $t=(gug)^{-1}\in T$. This shows that $T$ is normal. Now clearly, $\overline{U}=\overline{T}$. Let $x\in \overline{T}$ , then $x=t_1t_2...t_n$ for some $t_1,...,t_n\in T$. For any $g\in G$, we have:

$$gxg^{-1}=g(t_1t_2...t_n)g^{-1}=(gt_1g^{-1})(gt_2g^{-1})...(gt_ng^{-1}) $$

Which shows that $\overline{U}=\overline{T}\lhd G$, because $gt_1g^{-1},gt_2g^{-1},...,gt_ng^{-1}\in T$

Answer 4

There's an explicit characterization of the smallest subgroup containing a given set: $$\hat U=\{u_1^{\varepsilon_1}\ldots u_n^{\varepsilon_n}\mid u_i\in U,\varepsilon_i\in\{-1,1\}\text{ and }n\ge0\text{ (not a mistake)}\}$$ Because it must contain all these elements and it's closed under all group operations (including the identity).

Now observe that for any $g\in G$ we have $$gu_1^{\varepsilon_1}\ldots u_n^{\varepsilon_n}g^{-1}=gu_1^{\varepsilon_1}g^{-1}\ldots gu_n^{\varepsilon_n}g^{-1}=(gu_1g^{-1})^{\varepsilon_1}\ldots(gu_ng^{-1})^{\varepsilon_n}={u'_1}^{\varepsilon_1}\ldots{u'_n}^{\varepsilon_n}$$ So $\hat U$ is a normal subgroup of $G$.