# Finite direct power-closed characteristic is quotient-transitive

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This article gives the statement, and possibly proof, of a subgroup property (i.e., finite direct power-closed characteristic subgroup) satisfying a subgroup metaproperty (i.e., quotient-transitive subgroup property)
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## Statement

### Statement with symbols

Suppose $G$ is a group and $H$ and $K$ are subgroups with $H \le K \le G$. Suppose $H$ is a finite direct power-closed characteristic subgroup of $G$. Since characteristic implies normal, we can talk of the quotient group $G/H$. Suppose, further, that $K/H$ is a finite direct power-closed characteristic subgroup of $G/H$. Then, $K$ is also a finite direct power-closed characteristic subgroup of $G$.

## Facts used

1. Characteristicity is quotient-transitive: If $A \le B \le C$ with $A$ characteristic in $C$ and $B/A$ characteristic in $C/A$, then $B$ is characteristic in $C$.
2. (no link): The fact that $(K/H)^n \cong K^n/H^n$ and $(G/H)^n \cong G^n/H^n$, and the natural embedding from $(K/H)^n$ to $(G/H)^n$ coincides via these isomorphisms with the natural embedding from $K^n/H^n$ to $G^n/H^n$. where ${}^n$ stands for the $n^{th}$ direct power.

## Proof

Given: A group $G$, a finite direct power-closed characteristic subgroup $H$ of $G$. A subgroup $K$ of $G$ containing $H$ such that $K/H$ is a finite direct power-closed characteristic subgroup of $G/H$. $n$ is a natural number.

To prove: $K^n$ is a characteristic subgroup of $G^n$ under the natural embedding.

Proof:

Step no. Assertion Definitions used Facts used Given data used Previous steps used Explanation
1 $(K/H)^n$ is characteristic in $(G/H)^n$ finite direct power-closed characteristic subgroup -- $K/H$ is finite direct power-closed characteristic in $G/H$ -- Piece together definition and given data.
2 $K^n/H^n$ is characteristic in $G^n/H^n$ -- fact (2) -- step (1) Using the natural identification indicated by fact (2), step (2) becomes a reformulation of step (1).
3 $H^n$ is characteristic in $G^n$ finite direct power-closed characteristic subgroup -- $H$ is finite direct power-closed characteristic in $G$ -- Piece together definition and given data.
4 $K^n$ is characteristic in $G^n$ -- fact (1) -- steps (2) and (3) Piece together. [SHOW MORE]