Group action

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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 $(G,*)$ on a set $X$ is a mapping:

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

Notations for $g\cdot x$ include $gx$ and ${}^gx$

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

Notations for $x\cdot g$ include $xg$ and $x^g$

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