# Category:Group Homomorphisms

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This category contains results about Group Homomorphisms.

Definitions specific to this category can be found in Definitions/Group Homomorphisms.

Let $\struct {G, \circ}$ and $\struct {H, *}$ be groups.

Let $\phi: G \to H$ be a mapping such that $\circ$ has the morphism property under $\phi$.

That is, $\forall a, b \in G$:

- $\map \phi {a \circ b} = \map \phi a * \map \phi b$

Then $\phi: \struct {G, \circ} \to \struct {H, *}$ is a group homomorphism.

## Subcategories

This category has the following 15 subcategories, out of 15 total.

## Pages in category "Group Homomorphisms"

The following 36 pages are in this category, out of 36 total.

### C

### G

### H

### I

- Image of Canonical Injection is Kernel of Projection
- Image of Canonical Injection is Normal Subgroup
- Image of Group Homomorphism is Subgroup
- Image of Mapping from Group Element to Inner Automorphism is Inner Automorphism Group
- Inclusion Mapping on Subgroup is Homomorphism
- Induced Group Product is Homomorphism iff Commutative
- Induced Group Product is Homomorphism iff Commutative/Corollary
- Internal Group Direct Product of Normal Subgroups