Kernel is Normal Subgroup of Domain
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Theorem
Let $\phi$ be a group homomorphism.
Then the kernel of $\phi$ is a normal subgroup of the domain of $\phi$:
- $\map \ker \phi \lhd \Dom \phi$
Proof
Let $\phi: G_1 \to G_2$ be a group homomorphism, where the identities of $G_1$ and $G_2$ are $e_{G_1}$ and $e_{G_2}$ respectively.
By Kernel of Group Homomorphism is Subgroup:
- $\map \ker \phi \le \Dom \phi$
Let $k \in \map \ker \phi, x \in G_1$.
Then:
\(\ds \map \phi {x k x^{-1} }\) | \(=\) | \(\ds \map \phi x \map \phi k \paren {\map \phi x}^{-1}\) | Homomorphism to Group Preserves Inverses | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \phi x e_{G_2} \paren {\map \phi x}^{-1}\) | Definition of Kernel of Group Homomorphism | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \phi x \paren {\map \phi x}^{-1}\) | Definition of Identity Element | |||||||||||
\(\ds \) | \(=\) | \(\ds e_{G_2}\) | Definition of Inverse Element |
So:
- $x k x^{-1} \in \map \ker \phi$
This is true for all $k \in \map \ker \phi$ and $x \in G_1$.
From Subgroup is Normal iff Contains Conjugate Elements, it follows that $\map \ker \phi$ is a normal subgroup of $G_1$.
$\blacksquare$
Also see
Sources
- 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Algebraic Concepts
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 7.4$. Kernel and image
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 12$: Homomorphisms: Theorem $12.6$
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.10$: Theorem $22 \ (1)$
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Morphisms
- 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): $\S 2.2$: Homomorphisms
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Group Homomorphism and Isomorphism: $\S 65$
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 7$: Homomorphisms and quotient algebras
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 49.4$ Normal subgroups
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $8$: The Homomorphism Theorem: Theorem $8.13: \ (1)$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): kernel
- 2002: John B. Fraleigh: A First Course in Abstract Algebra (7th ed.): Chapter $13$: Corollary $13.20$
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): kernel