Finite Lie Group Names (awaiting review)

We describe Finite Groups of Lie Type (look at the Magma documentation here):

• $\mathrm{GL}(n,q)$ denotes the general linear group of degree $n$ over the finite field of order $q$.
• $\mathrm{GO}(n,q)$ denotes the general orthogonal group of degree $n$ over the finite field of order $q$.
• $\mathrm{GU}(n,q)$ denotes the general unitary group of degree $n$ over the finite field of order $q$.
• $\mathrm{SL}(n,q)$ denotes the special linear group of degree $n$ over the finite field of order $q$.
• $\mathrm{Sp}(n,q)$ denotes the symplectic group of degree $n$ over the finite field of order $q$.
• $\mathrm{SO}(n,q)$ denotes the special orthogonal group of degree $n$ over the finite field of order $q$.
• $\mathrm{SU}(n,q)$ denotes the special unitary group of degree $n$ over the finite field of order $q$.
• $\mathrm{Spin}(n,q)$ denotes the spin group of degree $n$ over the finite field of order $q$. It is a Lie group whose underlying manifold is the double cover of the special orthogonal group $\mathrm{SO}(n,q)$.

• Conformal groups:
  • $\mathrm{CO}(n,q)$ denotes the conformal orthogonal group of degree $n$ over the finite field of order $q$ consists of the product of the orthogonal group $\mathrm{O}(n,q)$ with the group of dilations. $\mathrm{CSO}(n,q)$ is defined in similar manner.
  • $\mathrm{CSp}(n,q)$ denotes the conformal symplectic group of degree $n$ over the finite field of order $q$. It preserves the standard alternating form up to a non-zero scalar multiple.
  • $\mathrm{CU}(n,q)$ denotes the conformal symplectic group of degree $n$ over the finite field of order $q$. It preserves the standard hermitian form up to a non-zero scalar multiple.

• Projective groups: $\mathrm{PG}(n,q)$ denotes the projective version of the group $G(n,q)$, which is the quotient of $G(n,q)$ by its center.

• Automorphism Groups:
  • $\mathrm{P\Gamma L}(n,q)$ denotes the automorphism group of $\mathrm{PGL}(n,q)$.
  • $\mathrm{P\Sigma L}(n,q)$ denotes the automorphism group of $\mathrm{PSL}(n,q)$.
  • $\mathrm{P\Gamma U}(n,q)$ denotes the automorphism group of $\mathrm{PGU}(n,q)$.
  • $\mathrm{P\Sigma Sp}(n,q)$ denotes the automorphism group of $\mathrm{PSp}(n,q)$.

• When the degree $n$ is even, there are two inequivalent non-degenerate quadratic forms giving rise to two non-isomorphic orthogonal groups $\mathrm{GO}^{+}(n,q)$ and $\mathrm{GO}^{-}(n,q)$ (see this wiki page).
  
  $G^{+}(n,q)$ and $G^{-}(n,q)$ are defined similarly for $G=\mathrm{CO}, \mathrm{CSO}, \mathrm{SO}, \mathrm{PSO}, \mathrm{PGO}, \mathrm{Spin}$.

• For $n\ge 3$ odd, $\Omega(n,q)$ is the subgroup of $\mathrm{SO}(n,q)$ such that $\mathrm{P}\Omega(n,q)$ is simple. If $q$ is odd, the group $\Omega(n,q)$ is the kernel of the spinor norm map on $\mathrm{SO}(n,q)$; if $q$ is even, it is the kernel of the Dickson invariant.
  
  For $n\ge 2$ even and $q$ is odd, $\Omega^{+}(n,q)$ (resp. $\Omega^{-}(n,q)$) is the kernel of the spinor norm map on $\mathrm{SO}^{+}(n,q)$ (resp. $\mathrm{SO}^{-}(n,q)$); if q is even, it is the kernel of the Dickson invariant. $\mathrm{P}\Omega^{+}(n,q)$ and $\mathrm{P}\Omega^{-}(n,q)$ are defined in a similar manner.

• Affine Groups:
  • $\mathrm{AGL}(n,q)$ denotes the affine general linear group of degree $n$ over the finite field of order $q$.
  • $\mathrm{ASL}(n,q)$ denotes the the affine special linear group of degree $n$ over the finite field of order $q$.
  • $\mathrm{A\Sigma Sp}(n,q)$ denotes the affine special symplectic group of degree $n$ over the finite field of order $q$.
  • $\mathrm{A\Gamma L}(n,q)$, $\mathrm{A\Sigma L}(n,q)$ and $\mathrm{ASp}(n,q)$ are defined in a similar manner.

• Chevalley groups:
  • $\mathrm{A_n}(q)$ is the special linear group $\mathrm{SL}(n+1,q)$.
  • $\mathrm{B_n}(q)$ is the orthogonal group $\Omega(2n+1,q)$.
  • $\mathrm{C_n}(q)$ is the symplectic group $\mathrm{Sp}(2n,q)$.
  • $\mathrm{D_n}(q)$ is the orthogonal group $\Omega^{+}(2n,q)$.
  • $\mathrm{E}(7,q)$, $\mathrm{E}(8,q)$, $\mathrm{F}(4,q)$, $\mathrm{G}(2,q)$ are the exceptional groups.