Definition of a Normal Subgroup

A subgroup $N$ of a group $G$ is called a normal subgroup if it is invariant under conjugation by any element of $G$. This means that for every element $g \in G$ and every element $n \in N$, the element $gng^{-1}$ is still in $N$. Formally, $N$ is normal in $G$ if: $gNg^{-1} = N \quad \text{for all } g \in G.$ This is often denoted as $N \triangleleft G$.

Proof that $HN$ is a Normal Subgroup of $H$

Let $H$ be a subgroup of $G$, and let $N$ be a normal subgroup of $G$. We want to show that the set $HN$ is a normal subgroup of $H$.

1. $HN$ is a Subgroup of $G$

First, note that $HN = \{ hn \mid h \in H, n \in N \}$. To prove that $HN$ is a subgroup of $G$, we need to show two things:

• Closure: If $x, y \in HN$, then $xy \in HN$.

  • Suppose $x = h_1n_1$ and $y = h_2n_2$ for some $h_1, h_2 \in H$ and $n_1, n_2 \in N$.
  • Then, $xy = (h_1n_1)(h_2n_2) = h_1(n_1h_2)n_2$.
  • Since $N$ is normal in $G$, $n_1h_2 = h_2n_1'$ for some $n_1' \in N$. Thus, $xy = h_1h_2n_1'n_2$, which is of the form $h'n'$ where $h' = h_1h_2 \in H$ and $n' = n_1'n_2 \in N$.
  • Therefore, $xy \in HN$, proving closure.

• Inverses: If $x \in HN$, then $x^{-1} \in HN$.

  • Suppose $x = hn$ for some $h \in H$ and $n \in N$.
  • Then $x^{-1} = (hn)^{-1} = n^{-1}h^{-1}$.
  • Again, since $N$ is normal in $G$, $n^{-1}h^{-1} = h^{-1}n'$ for some $n' \in N$.
  • Thus, $x^{-1} \in HN$, proving that $HN$ is a subgroup.

2. $HN$ is Normal in $H$

Now, we must show that $HN$ is a normal subgroup of $H$, meaning $h(HN)h^{-1} \subseteq HN$ for all $h \in H$.

• Take any $h_0 \in H$ and any element $hn \in HN$ where $h \in H$ and $n \in N$.
• Consider $h_0(hn)h_0^{-1} = (h_0h)n(h_0^{-1})$.
• Since $N$ is normal in $G$, $h_0nh_0^{-1} \in N$.
• Therefore, $h_0(hn)h_0^{-1} = (h_0h)n'$ for some $n' \in N$, and $h_0h \in H$.
• This shows that $h_0(hn)h_0^{-1} \in HN$.

Since $h_0$ and $hn$ were arbitrary, we have $h_0(HN)h_0^{-1} \subseteq HN$ for all $h_0 \in H$, meaning $HN$ is normal in $H$.

Conclusion

The set $HN$, where $H$ is a subgroup of $G$ and $N$ is a normal subgroup of $G$, is indeed a normal subgroup of $H$.

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Would you like more details or have any further questions? Here are five related questions to explore further:

1. What is an example of a normal subgroup in a familiar group like $\mathbb{Z}/n\mathbb{Z}$ or $S_n$?
2. How does the concept of a normal subgroup relate to the construction of quotient groups?
3. Can the intersection of two normal subgroups be non-normal? Why or why not?
4. What are the properties of the cosets of a normal subgroup?
5. How does the isomorphism theorem depend on the normality of subgroups?

Tip: When working with subgroups, remember that normality is essential for defining quotient groups, which are central to many areas of group theory.