Lagmental Vicfred

Cayley's Theorem Embeds Every Group into a Symmetric Group by Vicfred

tags: mathematics, permutation groups, group actions, conjugacy

Left multiplication realises an abstract group faithfully as permutations of its own underlying set. The formulas are more useful when each symbol has a job rather than merely decorating the theorem.

Notation

A permutation in \(S_n\) is best read through its disjoint cycle type. Group actions then translate algebra into orbits \(Gx\), stabilisers \(G_x\), and fixed-point counts.

$$ \lambda_g:G\to G,\qquad\lambda_g(x)=gx $$

A reliable calculation names domain and codomain. The notation \(\mathsf{data}\mapsto\mathsf{claim}\) is harmless only after both \(\operatorname{dom}\) and \(\operatorname{cod}\) have been fixed.

$$ \lambda:G\hookrightarrow\operatorname{Sym}(G) $$

Stress the formula

Now evaluate one representative case. The result should agree with the structural law above, but it is obtained without assuming the conclusion.

$$ \begin{aligned}\lambda_{gh}(x)&=(gh)x=g(hx)=(\lambda_g\circ\lambda_h)(x),\\\ker\lambda&=\{e\}.\end{aligned} $$

Interpretation

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

$$ \begin{aligned} \mathsf{D}\;&:\quad \lambda_g:G\to G,\qquad\lambda_g(x)=gx,\\[5pt] \mathsf{C}\;&:\quad \lambda:G\hookrightarrow\operatorname{Sym}(G). \end{aligned} $$

Limit of the argument

Cycle notation suppresses fixed points, so the ambient symmetric group still matters. The cycle \((1\,2\,3)\) in \(S_3\) and in \(S_8\) has different centralisers.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \lambda:G\hookrightarrow\operatorname{Sym}(G) \end{gathered}} $$

The notation is dense, but it is doing honest work: every delimiter records scope and every index records dependence. Removing one should require a mathematical reason.

This article was posted on Thu 23 June 2022. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.