Group action

Stub grade: A

This page is a stub

This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:

Needs fleshing out and neatening up, I'd like to introduce right group actions in a different way to left, however in my current attempt they're the same length!

Defintion

A (left) group action of a group [ilmath](G,*)[/ilmath] on a set [ilmath]X[/ilmath] is a mapping:

• [ilmath](\cdot):G\times X\rightarrow X[/ilmath] defined by [ilmath](\cdot):(g,x)\mapsto g\cdot x[/ilmath] such that:
  • [ilmath]\forall x\in X[1\cdot x=x][/ilmath] (where [ilmath]1[/ilmath] is the identity element of [ilmath](G,*)[/ilmath] group) and
  • [ilmath]\forall g,h\in G\ \forall x\in X[g\cdot(h\cdot x)=(g*h)\cdot x][/ilmath]

Notations for [ilmath]g\cdot x[/ilmath] include [ilmath]gx[/ilmath] and [ilmath]{}^gx[/ilmath]

A right group action is almost exactly the same, just the other way around; defined by [ilmath](\cdot):X\times G\rightarrow X[/ilmath] given by [ilmath](\cdot):(x,g)\mapsto x\cdot g[/ilmath] which must satisfy [ilmath]\forall x\in X[x\cdot 1=x][/ilmath] and [ilmath]\forall g,h\in G\ \forall x\in X[(x\cdot g)\cdot h=x\cdot(g*h)][/ilmath].

Notations for [ilmath]x\cdot g[/ilmath] include [ilmath]xg[/ilmath] and [ilmath]x^g[/ilmath]

See also

• Examples:
  • Every group acts on itself by multiplication
  • Every subgroup of a group acts on the group by multiplication
  • The symmetric group on a set acts on the set by evaluation
• Any group action can be thought of as a permutation