Lagmental Vicfred

The First Isomorphism Theorem Turns Fibres into Cosets by Vicfred

tags: mathematics, group theory, homomorphisms, quotients

Every fibre of a group homomorphism is a coset of the kernel, and the image is the resulting quotient. The formulas are more useful when each symbol has a job rather than merely decorating the theorem.

Set-up

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.

$$ \varphi:G\longrightarrow H,\qquad K=\ker\varphi $$

I keep the defining relation \(\mathsf D\) above the derived relation \(\mathsf C\). This exposes whether cancellation used \(x\ne0\) and whether the conclusion is canonical.

$$ G/K\cong\operatorname{im}\varphi $$

The calculation

Here is a concrete symbolic test. Reading it from left to right reveals which transformation is reversible and which is only an implication.

$$ \begin{aligned}\varphi(g_1)=\varphi(g_2)&\Longleftrightarrow\varphi(g_2^{-1}g_1)=e\\&\Longleftrightarrow g_1K=g_2K.\end{aligned} $$

What survives abstraction

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

$$ \begin{aligned} \mathsf{D}\;&:\quad \varphi:G\longrightarrow H,\qquad K=\ker\varphi,\\[5pt] \mathsf{C}\;&:\quad G/K\cong\operatorname{im}\varphi. \end{aligned} $$

The boundary

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.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] G/K\cong\operatorname{im}\varphi \end{gathered}} $$

The result is compact enough to reuse without pretending that the caveat has disappeared. The worked line remains the quickest consistency check.

This article was posted on Wed 03 January 2018. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.