Lagmental Vicfred

A Presentation Solves a Universal Mapping Problem by Vicfred

tags: mathematics, semidirect products, presentations, extensions

A map from a presented group is determined by images of generators that satisfy every defining relation. The example is deliberately concrete; it is a test of the statement, not a substitute for it.

Statement

A semidirect product combines groups \(N\) and \(H\) after choosing an action \(\alpha:H\to\operatorname{Aut}(N)\). A presentation records generators and relations but may conceal the size of the group.

$$ G=\langle X\mid R\rangle=F(X)/\langle\!\langle R\rangle\!\rangle $$

The first display fixes the mathematical data. I label it \(\mathsf{data}\) mentally, while the next is the \(\mathsf{claim}\); the bridge between them is the displayed \(\Longrightarrow\), not an automatic implication.

$$ \operatorname{Hom}(G,H)\cong\{f:X\to H:f(r)=e\ \forall r\in R\} $$

Worked algebra

The following line is the smallest calculation that still exercises the mechanism. It keeps nested delimiters and the order of operations explicit.

$$ \begin{gathered}G=\langle a,b\mid a^2=b^3=(ab)^5=e\rangle,\\ A^2=B^3=(AB)^5=I\Longrightarrow\exists!\,\varphi:G\to\operatorname{GL}(V).\end{gathered} $$

Conceptual compression

The compact alignment is a local map of the argument: assumptions on the first row, consequence on the second. Any generalisation must preserve that dependency.

$$ \begin{aligned} \mathsf{D}\;&:\quad G=\langle X\mid R\rangle=F(X)/\langle\!\langle R\rangle\!\rangle,\\[5pt] \mathsf{C}\;&:\quad \operatorname{Hom}(G,H)\cong\{f:X\to H:f(r)=e\ \forall r\in R\}. \end{aligned} $$

Caveat

Changing the action \(\alpha\) can change the group even when \(N\) and \(H\) remain fixed. The direct product is only the special case where the action is trivial.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \operatorname{Hom}(G,H)\cong\{f:X\to H:f(r)=e\ \forall r\in R\} \end{gathered}} $$

I would use the boxed line as a reference later, while returning to the full display whenever a hypothesis becomes uncertain. That division keeps compression from becoming ambiguity.

This article was posted on Wed 17 February 2016. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.