Lagmental Vicfred

The Correspondence Theorem Organises Subgroups above a Kernel by Vicfred

tags: mathematics, group theory, homomorphisms, quotients

Subgroups of a quotient correspond to subgroups upstairs that contain the normal subgroup being collapsed. The example is deliberately concrete; it is a test of the statement, not a substitute for it.

The data

A homomorphism \(\varphi:G\to H\) packages a comparison of operations. Its kernel \(\ker\varphi\) measures collapse, while its image \(\operatorname{im}\varphi\) records the part of \(H\) actually reached.

$$ \pi:G\to G/N,\qquad N\trianglelefteq G $$

The formulas should not be merged too early. The datum \(\mathsf D\), the conclusion \(\mathsf C\), and the bridge \(\Longrightarrow\) have three different logical jobs.

$$ \{K:N\le K\le G\}\longleftrightarrow\{L:L\le G/N\} $$

Derivation

An explicit case prevents the notation from becoming ceremonial. Every subscript and superscript in the display contributes to the value.

$$ \begin{aligned}K&\longmapsto K/N,\\L&\longmapsto\pi^{-1}(L),\\[2pt][G:K]&=[G/N:K/N].\end{aligned} $$

Invariant content

The invariant statement is the one that does not depend on a convenient choice of coordinates, representatives, basis, or enumeration.

$$ \begin{aligned} \mathsf{D}\;&:\quad \pi:G\to G/N,\qquad N\trianglelefteq G,\\[5pt] \mathsf{C}\;&:\quad \{K:N\le K\le G\}\longleftrightarrow\{L:L\le G/N\}. \end{aligned} $$

Scope

The quotient notation \(G/N\) is legal only for \(N\trianglelefteq G\). A set of cosets may exist without inheriting a well-defined group multiplication.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \{K:N\le K\le G\}\longleftrightarrow\{L:L\le G/N\} \end{gathered}} $$

A symbolic summary is trustworthy only because the example and limitation remain visible beside it. The box compresses the conclusion without hiding its origin.

This article was posted on Wed 22 February 2012. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.