Lagmental Vicfred

The Multiplication Law in a Semidirect Product by Vicfred

tags: mathematics, semidirect products, presentations, extensions

The action twists the second coordinate when pairs in N semidirect H are multiplied. The point is to make the formal expression readable enough to audit line by line.

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.

$$ \alpha:H\to\operatorname{Aut}(N),\qquad N\rtimes_\alpha H=N\times H $$

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.

$$ (n,h)(n',h')=(n\,\alpha_h(n'),hh') $$

Worked algebra

The middle display is intentionally dense: it is where signs, bounds, multiplicities, or normalising factors are most likely to be lost.

$$ \begin{aligned}(n,h)^{-1}&=\bigl(\alpha_{h^{-1}}(n^{-1}),h^{-1}\bigr),\\(e_N,e_H)&\text{ is the identity.}\end{aligned} $$

Conceptual compression

A good test for understanding is to change the presentation while keeping the invariant fixed. The aligned form makes that comparison unusually easy.

$$ \begin{aligned} \mathsf{D}\;&:\quad \alpha:H\to\operatorname{Aut}(N),\qquad N\rtimes_\alpha H=N\times H,\\[5pt] \mathsf{C}\;&:\quad (n,h)(n',h')=(n\,\alpha_h(n'),hh'). \end{aligned} $$

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.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] (n,h)(n',h')=(n\,\alpha_h(n'),hh') \end{gathered}} $$

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 Thu 02 July 2020. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.