Group

Proposition

(1)

\[(aga^{-1})^{-1} = ag^{-1}a^{-1} .\]

proof

$\Box$

Definition

• $G$,
  • group
• $S \subseteq G$,
  • subgroup

\[aS := \{ ag \mid g \in S \} .\]

■

Definitoin

• $G$,
  • gorup
• $N \subseteq G$,
  • subgroup

$N$ is said to be a normal group if

\[\forall n \in N, g \in G, \quad gng^{-1} \in N.\]

■

Proposition

• $G$,
  • group
• $N \subseteq G$
  • normal subgroup
• $a \in G$,
• $n \in N$,

(1) $a^{-1}na \in N$.

(2) $aN = Na$

proof

(1)

By definition, $an^{-1}a^{-1} \in N$. $N$ is group so the inverse is contained in $N$. That is,

\[(an^{-1}a^{-1})^{-1} = a^{-1}na \in N .\]

(2)

\[an = ana^{-1}a\]

Here $ana^{-1} \in N$. Hence $an \in Na$. Similary,

\[na = aa^{-1}Na \in aN .\]

$\Box$

Definition commutator

• $G$,
  • group

\[g, h \in G, \quad [g, h] := g^{-1}h^{-1}gh .\]

$[g, h]$ is called commutator of $g, h$.

\[[G, G] := \{ [g_{1}, h_{1}] \cdots [g_{n}, h_{n}] \mid n \in \mathbb{N}, \ g_{i}, h_{i} \in G \} .\]

$[G, G]$ is called the commutator group of $G$.

■

Propositon

• $G$,
  • group
• $g, h \in G$,
• $a \in G$,

(1) $a[g, h]a^{-1} = [aga^{-1}, aha^{-1}]$

(2) $[G, G]$ is the smallest normal subgroup

proof

(1)

\[\begin{eqnarray} [aga^{-1}, aha^{-1}] & = & (aga^{-1})^{-1}(aha^{-1})^{-1}aga^{-1}aha^{-1} \nonumber \\ & = & ag^{-1}a^{-1}ah^{-1}a^{-1} aga^{-1}aha^{-1} \nonumber \\ & = & ag^{-1}h^{-1}gha^{-1} \nonumber \\ & = & a[g, h]a^{-1} \nonumber \end{eqnarray}\]

$\Box$

Reference