Subgroup of Abelian Group is Normal

Theorem

Every subgroup of an abelian group is normal.

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 \ \ $

$\ds y$

$\in$

$\ds H$

$\blacksquare$

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 $7.5$