# APS of groups

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

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

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## Contents

## Definition

An APS of groups is an APS over the category of groups. More specifically, an APS of groups is the following data:

- For each natural number , a group, denoted .
- For each ordered pair of natural numbers, a 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.

## Terminology

### Members and elements

For an APS the member is termed the member of the APS. A **member** of the APS is an object that is the member for some .

An **element** of the APS is an element of some member of the APS.

The **home** of an element of the APS is the member in which it lies. The **index** of a member is the for which it is the member.

### Block concatenation map

The maps are termed **block concatenation maps**.

### Ground member

The ground member of an APS is its first member.

## Other notions

### Homomorphism of APSes

`Further information: APS homomorphism`

Given APSes and , a homomorphism → associates, to each natural number , a map → , such that:

### Sub-APS notion

`Further information: sub-APS`

Given an APS , a sub-APS associates, to each , a subgroup of , such that the image of under lies inside .

When the APS of groups is injective, any sub-APS is also injective.

### Quotient APS notion

`Further information: quotient APS`

A quotient APS is the image of an APS in a homomorphism that is surjective at each member.

### Normal sub-APSes, kernels and images

Given a homomorphism of APSes of groups, the kernels of the individual homomorphisms for a sub-APS of the domain APS, and the images of the individual homomorphism form a sub-APS of the range APS. The image is thus a quotient APS.

Further, we have the following result: a sub-APS of an APS of groups occurs as the kernel of an APS homomorphism if and only if every member of it is a normal subgroup of the corresponding member of the whole APS. A sub-APS satisfying either of these equivalent conditions is termed a normal sub-APS.

This parallels the group theory result that a subgroup of a group occurs as the kernel of a group homomorphism if and only if it is normal.

## Properties

### Injectivity

An APS of groups is termed injective, or an IAPS of groups, if every block concatenation map is injective. For an IAPS of groups, we usually also assume the condition of refinability.

### Commutativity

Very few APSes of groups are commutative. Note that a commutative APS cannot also be injective.

### Padding-injectivity

Most APSes of groups that we encounter satisfy the condition of being padding-injective.