# Notes - Groups TT23, Group actions

### Flashcards

Define a left action of a group $G$ on a set $S$.

A map

\[\rho : G\times S \to S\]such that

- $\rho(e, s) = s$ for all $s \in S$
- $\rho(g, \rho(h, s)) = \rho(gh, s)$

Can you formulate conjugation as a group action?

$G$ acts on itself by

\[g \cdot h = ghg^{-1}\]