# Endomorphism kernel is quotient-transitive

This article gives the statement, and possibly proof, of a subgroup property (i.e., endomorphism kernel) satisfying a subgroup metaproperty (i.e., quotient-transitive subgroup property)
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## Statement

Suppose $H \le K \le G$ are groups such that $H$ is an endomorphism kernel in $G$ and the quotient group $K/H$ is an endomorphism kernel in $G/H$. (Note that it makes sense to take quotients because endomorphism kernel implies normal). Then, $K$ is an endomorphism kernel in $G$.

## Definitions used

We use the following definition of endomorphism kernel: a normal subgroup $A$ of a group $B$ is an endomorphism kernel if there exists a subgroup of $B$ isomorphic to the quotient group $B/A$.

## Facts used

1. Third isomorphism theorem: This basically tells us that $(G/H)/(K/H) \cong G/K$.

## Proof

Given: Groups $H \le K \le G$ such that $H$ is an endomorphism kernel in $G$ and $K/H$ is an endomorphism kernel in $G/H$.

To prove: $K$ is an endomorphism kernel in $G$.

Proof:

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1 Let $L$ be a subgroup of $G$ isomorphic to $G/H$ and let $\alpha:G/H \to L$ be an isomorphism between them. $H$ is an endomorphism kernel in $G$ Definition-direct
2 Let $M$ be a subgroup of $G/H$ isomorphic to $(G/H)/(K/H)$. $K/H$ is an endomorphism kernel in $G/H$. Definition-direct
3 $(G/H)/(K/H)$ is isomorphic to $G/K$. Fact (1) $H \le K \le G$, $H$ normal in $G$, $K/H$ normal in $G/H$ (normality follows from being endomorphism kernels) Fact-direct
4 $M$ is a subgroup of $G/H$ isomorphic to $G/K$. Steps (2), (3) step-combination direct
5 $\alpha(M)$ is a subgroup of $L$ (and hence of $G$) isomorphic to $M$ Steps (1), (2) $\alpha:G/H \to L$ is an isomorphism, hence it maps subgroups to subgroups isomorphic to them. Apply this to $M$.
6 $\alpha(M)$ is a subgroup of $G$ isomorphic to $G/K$. Steps (4), (5) step-combination direct.
7 $K$ is an endomorphism kernel in $G$. Step (6) definition-direct.