Subgroup of Abelian Group is Normal
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Theorem
Every subgroup of an abelian group is normal.
Proof
Let $G$ be an abelian group.
Let $H \le G$ be a subgroup of $G$.
Then for all $a \in G$:
\(\ds y\) | \(\in\) | \(\ds H^a\) | where $H^a$ is the conjugate of $H$ by $a$ | |||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds a y a^{-1}\) | \(\in\) | \(\ds H\) | Definition of Conjugate of Group Subset | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds y\) | \(\in\) | \(\ds H\) | because $a y a^{-1} = y$ as $G$ is abelian |
$\blacksquare$
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 6.6$. Normal subgroups
- 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 $36$
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Morphisms
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Conjugacy, Normal Subgroups, and Quotient Groups: $\S 46$
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 5$: Groups $\text{I}$: Subgroups
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 49.6$ Normal subgroups
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $7$: Normal subgroups and quotient groups: Example $7.5$