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Problem 87 (Some Normal Subgroups)

Prove the following facts.

  1. If $G$ is Abelian, then every subgroup of $G$ is normal.
  2. The center $Z(G)$ of any group $G$ is always a normal subgroup of $G$.

Solution

Let $G$ be a group.

  1. Suppose $G$ is Abelian. Let $H\leq G$. We will show that $H\trianglelefteq G$. Let $h\in H$ and $a\in G$. By definition of coset we know that $ha\in Ha$ and $ah\in aH$. Because $H\subseteq G$ we know that $h\in G$. It follows then that $ha=ah$ because $h\in G$ and $G$ is Abelian. Since this is true for every $h\in H$ and $a\in G$ we know that $H\trianglelefteq G. \square$
  2. Let $F=Z(G)=\{ x\in G | gx=xg \text{ for all } g\in G\}$. We will show that $F\trianglelefteq G$. Pick $y\in F$ and $b\in G$. We know that $yb=by$ by definition. Because this is true for every $y\in F$ we know that $Fb=bF$. This is true for every $b\in G$ by definition of the center of a group. Hence, $F\trianglelefteq G. \square$
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