Image of Group Homomorphism is Subgroup

Theorem

Let $\phi: G_1 \to G_2$ be a group homomorphism.

Then:

$\Img \phi \le G_2$

where $\le$ denotes the relation of being a subgroup.

Proof

This is a special case of Group Homomorphism Preserves Subgroups, where we set $H = G_1$.

$\blacksquare$

Sources

• 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 $2$: Group Homomorphism and Isomorphism: $\S 66$
• 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 7$: Homomorphisms and quotient algebras
• 1974: Thomas W. Hungerford: Algebra ... (previous) ... (next): $\S 1.2$
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 47.4$ Homomorphisms and their elementary properties
• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $8$: The Homomorphism Theorem: Theorem $8.13: \ (2)$