Affine, almost simple, diagonal, product action, and twisted wreath types

For a finite primitive permutation group $G\le \mathrm{Sym}(\mathrm{\Omega })$, the five coarse O'Nan-Scott types used on this page are:

• Affine type: the socle is the unique minimal normal subgroup, it is abelian and regular, and A unique abelian minimal normal subgroup gives affine type identifies it with a finite vector space.
• Almost simple type: the socle is a nonabelian simple group and the whole group lies between that socle and its full automorphism group in the sense of Almost simple finite groups.
• Diagonal type: the socle is a direct product $T^k$, with $k\ge 2$, of isomorphic nonabelian simple groups, and the action is the standard diagonal action on a coset space of a diagonal subgroup.
• Product action type: after identifying $\mathrm{\Omega }$ with ${\mathrm{\Delta }}^{\mathrm{\ell }}$ for some $\mathrm{\ell }\ge 2$, there is a primitive group $H$ on $\mathrm{\Delta }$ of almost simple or diagonal type, with $N=\mathrm{Soc}(H)$, such that $N^{\mathrm{\ell }}=\mathrm{Soc}(G)\le G\le H\wr K,$ where $K\le S_{\mathrm{\ell }}$ is the transitive group induced by $G$ on the coordinates and the wreath product has its product action. If $(h_1,\dots ,h_{\mathrm{\ell }};k)\in H^{\mathrm{\ell }}⋊K$, its product action is $({\delta }_1,\dots ,{\delta }_{\mathrm{\ell }})⟼({\delta }_{k^{-1}(1)}{h}_{k^{-1}(1)},\dots ,{\delta }_{k^{-1}(\mathrm{\ell })}{h}_{k^{-1}(\mathrm{\ell })})\mathrm{.}$
• Twisted wreath type: $G$ is permutation equivalent to the following group on $B$, and this action is primitive. Take a finite nonabelian simple group $T$, a faithful transitive permutation group $P\le S_k$, $k\ge 2$, its point stabilizer $Q=P_1$, and a homomorphism $\phi :Q\to \mathrm{Aut}(T)$ whose image contains $\mathrm{Inn}(T)$. With automorphisms composed as left operators, set $B=\{f:P\to T:f(xq)=\phi (q{)}^{-1}(f(x))\text{ for every }x\in P,q\in Q\}\mathrm{.}$ Multiplication in $B$ is pointwise. Define ${\alpha }_p(f)(x)=f(p^{-1}x)\mathrm{.}$ The twisted wreath product is $B{⋊}_{\alpha }P$, with the convention of The external semidirect product $N{⋊}_{\alpha }H$, acting on $B$ by $(b,p)\cdot c=b{\alpha }_p(c)\mathrm{.}$ Its socle is the unique minimal normal subgroup $B\cong T^k$, acting regularly; its degree is $\mathrm{\mid }T{\mathrm{\mid }}^k$. Primitivity is a required condition on these data, not a consequence of transitivity of $P$ alone.

The function construction is well defined: specifying values on one representative of each of the $k$ cosets $xQ$ determines a unique function, because $\phi (q_1{q}_2{)}^{-1}=\phi (q_2{)}^{-1}\phi (q_1{)}^{-1}$. Evaluation there identifies the pointwise group with $T^k$. The maps ${\alpha }_p$ preserve its defining condition and satisfy ${\alpha }_p{\alpha }_r={\alpha }_{pr}$, so The semidirect-product multiplication makes $N\times H$ a group applies and the displayed permutation formula respects multiplication. These are finite choices.

For the socle assertion, the normal subgroups of $T^k$ are products of its factors: commutating an element of a normal subgroup with one factor isolates that coordinate, and simplicity and the trivial centre of $T$ then give the entire factor whenever its projection is nontrivial. The transitive action of $P$ on the factors makes $B$ minimal normal. If an element $(b,p)$ centralizes $B$, then ${\alpha }_p$ is an inner automorphism of $B$ and hence fixes every factor. Faithfulness of $P$ forces $p=1$, and then $b$ is central in $B$, so $b=1$. Any distinct minimal normal subgroup would centralize $B$ (their commutator lies in their trivial intersection). Therefore no such subgroup exists.

These conventions implement LPS Section 1, type III(c), using left actions. Defining these types does not prove that every finite primitive group belongs to one of them, nor that a proof of that classification avoids CFSG.