CFSG

Theorem

Every finite simple group is isomorphic to one of the followings:

• a cyclic group ${\mathbb{Z}}_p$ of prime power $p$;
• an alternating group $A_n$ for $n⩾5$;
• a classical group with a prime power $q=p^a$:
  • linear groups: ${\mathrm{PSL}}_n(q)$ for $n⩾2$ except ${\mathrm{PSL}}_2(2)$ and ${\mathrm{PSL}}_2(3)$;
  • unitary groups: ${\mathrm{PSU}}_n(q)$ for $n⩾3$ except ${\mathrm{PSU}}_3(2)$;
  • symplectic groups: ${\mathrm{PSp}}_{2n}(q)$ for $n⩾2$ except ${\mathrm{PSp}}_4(2)$;
  • orthogonal groups: $\mathrm{P}{\Omega }_{2n}^{+}(q)$, $\mathrm{P}{\Omega }_{2n}^{-}(q)$ for $n⩾4$, and $\mathrm{P}{\Omega }_{2n+1}(q)$ for $n⩾3$ and $q$ odd;
• an exceptional group of Lie type:
  • $G_2(q),q⩾3;F_4(q);E_6(q);{}^2{E}_6(q);{}^{3}D(q);E_7(q);E_8(q)$, where $q$ is a prime power, or
  • ${}^2{B}_2(2^{2n+1}),n⩾1;{}^2{G}_2(3^{2n+1}),n⩾1;{}^2{F}_4(2^{2n+1}),n⩾1$, or
  • the Tits group ${}^2{F}_2(2{)}^{\prime }$;
• $26$ sporadic simple groups.