Trivial Subgroup is Normal
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Theorem
Let $\struct {G, \circ}$ be a group whose identity is $e$.
Then the trivial subgroup $\struct {\set e, \circ}$ of $G$ is a normal subgroup in $G$.
Proof
First, by Trivial Subgroup is Subgroup, $\struct {\set e, \circ}$ is a subgroup of $G$.
To show $\struct {\set e, \circ}$ is normal in $G$:
- $\forall a \in G: a \circ e \circ a^{-1} = a \circ a^{-1} = e$
Hence the result.
$\blacksquare$
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 6.6$. Normal subgroups: Example $123$
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.10$: Example $35$
- 1978: John S. Rose: A Course on Group Theory ... (previous) ... (next): $1$: Introduction to Finite Group Theory: $1.7$
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 49.1$ Normal subgroups
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $7$: Normal subgroups and quotient groups: Definition $7.3$: Remark