# Sub-(isomorph-normal characteristic) subgroup

## Contents

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## Definition

A subgroup $H$ of a group $G$ is termed a sub-(isomorph-normal characteristic) subgroup if there exists an ascending chain of subgroups:

$H = H_0 \le H_1 \le H_2 \le \dots \le H_n = G$

such that each $H_i$ is an isomorph-normal characteristic subgroup of $H_{i+1}$, i.e., $H_i$ is characteristic in $H_{i+1}$, and $H_i$ is isomorph-normal in $H_{i+1}$: every subgroup of $H_{i+1}$ isomorphic to $H_i$ is normal.

## Metaproperties

### Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity