Normal Subgroups

Subgroups and cosets

$G$: a group.

$H \leq G$: a subgroup of $G$.

$gH = \{gh \mid h \in H \}$: left coset of $H$ with respect to $g \in G$.

$Hg = \{hg \mid h \in H \}$: right coset of $H$ with respect to $g \in G$.

$L = \{gH \mid g \in G \}$: set of all left cosets of $H$.

$R = \{Hg \mid g \in G \}$: set of all right cosets of $H$.

Normal subgroup

$N$: a subgroup of $G$.

$gN = Ng \quad \forall , g \in G$: $N$ is a normal subgroup of $G$.

$N \trianglelefteq G$: notation for normality.

$G \trianglelefteq G$: normal subgroup of itself.

$\{ 1 \} \trianglelefteq G$: trivial subgroup is normal.

e.g. Not normal

$S_3 = \{ (), (1 \ 2), (1 \ 3), (2 \ 3), (1 \ 2 \ 3), (1 \ 3 \ 2) \}$: symmetric group.

$S_2 = \{ (), (1 \ 2) \}$: symmetric group.

$S_2 \leq S_3$: subgroup.

$(2 \ 3 ) S_2 = \{ (2 \ 3), (1 \ 2 \ 3) \}$: left coset.

$S_2 (2 \ 3 ) = \{ (2 \ 3), (1 \ 3 \ 2) \}$: right coset.

$S_2$ is not normal in $S_3$.

e.g. Normal

$S_3 = \{ (), (1 \ 2), (1 \ 3), (2 \ 3), (1 \ 2 \ 3), (1 \ 3 \ 2) \}$: symmetric group.

$A_3 = \{ (), (1 \ 2 \ 3), (1 \ 3 \ 2) \}$: alternating group.

$A_3 \leq S_3$: subgroup.

$(2 \ 3 ) A_3 = \{ (2 \ 3), (1 \ 2), (1 \ 3) \}$: left coset.

$A_3 (2 \ 3 ) = \{ (2 \ 3), (1 \ 3), (1 \ 2) \}$: right coset.

$A_3$ is normal in $S_3$ but must check the others.