Definition:Normal Subset
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Definition
Let $\struct {G, \circ}$ be a group.
Let $S \subseteq G$ be a general subset of $G$.
Then $S$ is a normal subset of $G$ if and only if:
Definition 1
- $\forall g \in G: g \circ S = S \circ g$
Definition 2
- $\forall g \in G: g \circ S \circ g^{-1} = S$
or, equivalently:
- $\forall g \in G: g^{-1} \circ S \circ g = S$
Definition 3
- $\forall g \in G: g \circ S \circ g^{-1} \subseteq S$
or, equivalently:
- $\forall g \in G: g^{-1} \circ S \circ g \subseteq S$
Definition 4
- $\forall g \in G: S \subseteq g \circ S \circ g^{-1}$
or, equivalently:
- $\forall g \in G: S \subseteq g^{-1} \circ S \circ g$
Definition 5
- $\forall x, y \in G: x \circ y \in S \implies y \circ x \in S$
Definition 6
- $\map {N_G} S = G$
where $\map {N_G} S$ denotes the normalizer of $S$ in $G$.
Definition 7
- $\forall g \in G: g \circ S \subseteq S \circ g$
or:
- $\forall g \in G: S \circ g \subseteq g \circ S$
Also see
- Results about normal subsets can be found here.