Throughout, \(G\) is a group, \(H\le G\) is a subgroup, \(g,h,x\in G\), and \(X\) is a set.
Subsets and subgroups
| Notation | Name | Definition |
|---|---|---|
| \(\mcp(X)\), \(2^X\) | power set of \(X\) | the set of subsets of \(X\), in bijection with \(\Hom_\Set(X,\ts{0,1})\) |
| \([g,h]\) | commutator of \(g\) and \(h\) | \(gh\inverseof{g} \inverseof{h}\) |
| \([G,H]\) | commutator subgroup of \(G\) and \(H\) | \(\gens{[g,h] \st g\in G,\ h\in H}\) |
| \(Z(G)\) | center of \(G\) | \(\ts{x\in G \st gx\inverseof{g} = x \text{ for all } g\in G}\) |
| \(C_G(x)\), \(Z(x)\) | centralizer of \(x\) | \(\ts{g\in G \st [g,x]=1}\) |
| \(C_G(H)\), \(Z_G(H)\) | centralizer of \(H\) | \(\ts{g\in G \st [g,h]=1 \text{ for all } h\in H} = \Intersect_{h\in H} C_G(h)\) |
| \([x]\), \(\Conj(x)\) | conjugacy class of \(x\) | \(\ts{gx\inverseof{g} \st g\in G}\subseteq G\) |
| \(N_G(H)\) | normalizer of \(H\) | \(\ts{g\in G \st gH\inverseof{g} = H}\le G\) |
| \(\Inn(G)\) | inner automorphisms | \(\ts{\varphi_g \st g\in G}\le\Aut(G)\), where \(\varphi_g(x) = gx\inverseof{g}\) |
| \(\Out(G)\) | outer automorphisms | \(\Aut(G)/\Inn(G)\) |
Group actions
For a group action of \(G\) on \(X\), given by a homomorphism \(\phi\colon G\to\Aut_\Set(X)\), and for \(x\in X\):
| Notation | Name | Definition |
|---|---|---|
| \(g\cdot x\) | action of \(g\) on \(x\) | \(\phi(g)(x)\) |
| \(\Orb(x)\), \(Gx\) | orbit of \(x\) | \(\ts{g\cdot x \st g\in G}\subseteq X\) |
| \(\Stab(x)\), \(\Stab_G(x)\), \(G_x\) | stabilizer of \(x\) | \(\ts{g\in G \st g\cdot x = x}\le G\) |
| \(X/G\) | set of orbits | \(\ts{\Orb(x) \st x\in X}\subseteq 2^X\) |
| \(\Fix(X)\), \(\Fix_G(X)\), \(X^G\) | set of fixed points | \(\ts{x\in X \st g\cdot x = x \text{ for all } g\in G}\subseteq X\) |