# Category:Group Direct Products

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This category contains results about Group Direct Products.

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

Let $\struct {G, \circ_1}$ and $\struct {H, \circ_2}$ be groups.

Let $G \times H: \set {\tuple {g, h}: g \in G, h \in H}$ be their cartesian product.

The **(external) direct product** of $\struct {G, \circ_1}$ and $\struct {H, \circ_2}$ is the group $\struct {G \times H, \circ}$ where the operation $\circ$ is defined as:

- $\tuple {g_1, h_1} \circ \tuple {g_2, h_2} = \tuple {g_1 \circ_1 g_2, h_1 \circ_2 h_2}$

This is usually referred to as the **group direct product** of $G$ and $H$.

## Subcategories

This category has the following 12 subcategories, out of 12 total.

### C

### D

### E

### G

### I

## Pages in category "Group Direct Products"

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

### A

### C

### D

- Direct Product of Central Subgroups
- Direct Product of Group Homomorphisms is Homomorphism
- Direct Product of Normal Subgroups is Normal
- Direct Product of Solvable Groups is Solvable
- Direct Product of Sylow p-Subgroups is Sylow p-Subgroup
- Direct Product of Unique Sylow p-Subgroups is Unique Sylow p-Subgroup

### E

### G

### I

- Identity of Group Direct Product
- Image of Canonical Injection is Kernel of Projection
- Image of Canonical Injection is Normal Subgroup
- Internal and External Group Direct Products are Isomorphic
- Intersection with Normal Subgroup is Normal/Examples/Subset Product of Intersection with Intersection
- Inverses in Group Direct Product