# IAPS of groups

From Groupprops

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This article gives a basic definition in the following area: APS theory

View other basic definitions in APS theory |View terms related to APS theory |View facts related to APS theory

This article defines the notion of group object in the category of IAPSs|View other types of group objects

## Definition

An **IAPS of groups** is an IAPS over the category of groups. More specifically an IAPS is the following data:

- For each natural number , a group denoted
- For each ordered pair of natural numbers, an injective homomorphism

Satisfying the following compatibility conditions:

For in respectively:

.

The above condition is termed an associativity condition.

We may assume as the trivial group and define and as trivial paddings.

Note that if we remove the condition of injectivity, we get an APS of groups.

## Constructions

### Sub-IAPS

`Further information: sub-IAPS of groups`

Let be an IAPS of groups. A **sub-IAPS** associated to every a subgroup of such that the image of under lies inside . Note that this is the same as a sub-APS of groups because the injectivity condition comes for free.

### Quotient IAPS

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