03-01. Definitions and examples

Definition. normal subgroup

• $G$,
  • group
• $N \subseteq G$,
  • subset
• $n \in N$,

• $\bar{g} \in G$,

• (1) $\bar{g}n\bar{g}^{-1}$ is called the conjugate of $n$ by $\bar{g}$.
• (2) $gNg^{-1}$ is accalled the conjugate of $N$ by $g$.
• (3) $g$ is said to normalize $N$ if $gNg^{-1} = N$.
• (4) $N$ is said to be normal if $gNg^{-1} = N$ for all $g \in G$.

If $N$ is normal, we write $N \trianglelefteq G$.

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Theorem 6.

• $G$,
  • group
• $N$,
  • subgroup

The following are euivalent:

• (1) $N \trianglelefteq G$,
• (2) $N_{G}(N) = G$,
• (3) $nN = Nn$ for all $g \in G$,
• (4)
• (5) $gNg^{-1}$

proof.

$\Box$