# Subgroup series

## Contents

## Definition

### Symbol-free definition

A **subgroup series** in a group is an order-preserving map from a totally ordered indexing set to the collection of subgroups of the group, ordered by inclusion.

### Definition with symbols

A **subgroup series** in a group is a map from a totally ordered indexing set to the collection of subgroups of , that sends in to , such that if , then is a subgroup of .

## Particular cases

### Well-ordered indexing set (for minimum)

When the indexing set is well-ordered, the subgroup series is termed an ascending series. For any well-ordered indexing set, every element is either a successor element or a limit element. If, further, we have the property that for a limit element, the corresponding subgroup is the union of the subgroups below it, we call the ascending series limit-tight.

### Well-ordered indexing set (for maximum)

When every subset has a maximum (that is, the indexing set is well-ordered with respect to maximum), the subgroup series is termed a descending series. If, further, we have the property that for a limit element, the corresponding subgroup is the intersection of the subgroups above it, we call the descending series limit-tight.

## Operations on subgroup series

### Concatenation of subgroup series

Consider a subgroup series in a group where the intersection of all members of the series is . Let be a subgroup series in . Then the concatenation of with is a subgroup series of which essentialyl concatenates the subgroup series.**PLACEHOLDER FOR INFORMATION TO BE FILLED IN**: [SHOW MORE]

### Intersection of subgroup series

Let and be two subgroup series in a group both indexed by the same indexing set . Then, the intersection of and , denoted , associated to each element the subgroup .

### Join of subgroup series

Let and be two subgroup series in a group both indexed by the same indexing set . Then, the intersection of and , denoted , associated to each element the subgroup .

## Maps from subgroup properties to subgroup series properties

### The successor condition map

Given a subgroup property , the successor condition map gives a subgroup series property that a subgroup series satisfies if and only if:

- For any member , satisfies as a subgroup of the intersection of all for .
- For any member , the subgroup generated by for satisfies the property as a subgroup of .

### The comparison condition map

Given a subgroup property , the comparison condition map gives the property of being a subgroup series where, given any subsets and of such that every element of is less than every element of the subgroup generated by all the subgroups corresponding to satisfies as a subgroup of the intersection of all the subgroups corresponding to .