Subgroup is Normal iff Contains Conjugate Elements

Theorem

Let $\struct {G, \circ}$ be a group whose identity is $e$.

Let $N$ be a subgroup of $G$.

Then $N$ is normal in $G$ if and only if:

$\text {(1)}: \quad$

$\ds \forall g \in G: \, $

$\ds n \in N$

$\iff$

$\ds g \circ n \circ g^{-1} \in N$

$\text {(2)}: \quad$

$\ds \forall g \in G: \, $

$\ds n \in N$

$\iff$

$\ds g^{-1} \circ n \circ g \in N$

Proof

By definition, a subgroup is normal in $G$ if and only if:

$\forall g \in G: g \circ N = N \circ g$

Necessary Condition

Suppose that $g \circ N = N \circ g$, by definition 1 of normality in $G$.

Let $n \in N$.

Then:

$\ds g \circ n$

$\in$

$\ds N \circ g$

Definition of Coset

$\ds \leadstoandfrom \ \ $

$\ds \exists n_1 \in N: \, $

$\ds g \circ n$

$=$

$\ds n_1 \circ g$

Definition of Coset

$\ds \leadstoandfrom \ \ $

$\ds g \circ n \circ g^{-1}$

$=$

$\ds n_1$

Division Laws for Groups

$\ds \leadstoandfrom \ \ $

$\ds g \circ n \circ g^{-1}$

$\in$

$\ds N$

Definition of $n_1$

$\Box$

Sufficient Condition

Suppose that:

$\forall g \in G: \paren {n \in N \iff g \circ n \circ g^{-1} \in N}$

Let $g \circ n \circ g^{-1} \in N$.

$\ds \exists n_1 \in N: \, $

$\ds g \circ n \circ g^{-1}$

$=$

$\ds n_1$

$\ds \leadsto \ \ $

$\ds g \circ n$

$=$

$\ds n_1 \circ g$

Division Laws for Groups

$\ds \leadsto \ \ $

$\ds g \circ n$

$\in$

$\ds N \circ g$

Definition of Coset

$\ds \leadsto \ \ $

$\ds g \circ N$

$\subseteq$

$\ds N \circ g$

Definition of Subset

Similarly:

$\ds \exists n_2 \in N: \, $

$\ds g \circ n \circ g^{-1}$

$=$

$\ds n_2$

$\ds \leadsto \ \ $

$\ds n \circ g^{-1}$

$=$

$\ds g^{-1} \circ n_2$

Division Laws for Groups

$\ds \leadsto \ \ $

$\ds n \circ g^{-1}$

$\in$

$\ds g^{-1} \circ N$

Definition of Coset

$\ds \leadsto \ \ $

$\ds N \circ g^{-1}$

$\subseteq$

$\ds g^{-1} \circ N$

Definition of Subset

As $g$ is arbitrary, then so is $g^{-1}$.

Thus:

$N \circ g \subseteq g \circ N$

By definition of set equality:

$g \circ N = N \circ g$

Also see

Equivalence of Definitions of Normal Subgroup

Sources

1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 6.6$. Normal subgroups
1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.10$: Theorem $26$
1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): $\text{II}$: Morphisms
1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 49.3$ Normal subgroups
1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): $\S 7$: Definition $7.3$