# Subgroups of arbitrary even ordinary depth

## TL;DR

This paper constructs specific examples of groups and subgroups demonstrating that even ordinary depth can be arbitrarily large, specifically for any positive integer n, resolving an open problem in group theory.

## Contribution

It provides the first known examples of subgroups with arbitrary even ordinary depth, answering a longstanding open question.

## Key findings

- Existence of groups with subgroups of arbitrary even depth
- Construction of examples for each positive integer n
- Resolution of the open problem on large even ordinary depth

## Abstract

We show that for each positive integer $n$, there are a group $G$ and a subgroup $H$ such that the ordinary depth is $d(H, G) = 2n$. This solves the open problem posed by Lars Kadison whether even ordinary depth larger than $6$ can occur.