Primitive group action

A transitive group action is primitive if the only blocks are singletons and the whole set; a block has every translate either equal to it or disjoint from it. Orbits of a normal subgroup form blocks. Hence a nontrivial normal subgroup in a faithful primitive action is transitive. Two-transitivity implies primitivity. This normal-subgroup observation is a hypothesis engine for Iwasawa's simplicity lemma.

If a primitive subgroup of $S_n$ contains a transposition, make a graph whose edges are the supports of its conjugate transpositions. Connected components form blocks for the group. An edge exists, so primitivity forces the graph to be connected. Its edge transpositions generate all of $S_n$, proving that the original group is $S_n$.

Every nontrivial normal subgroup of a faithful primitive group of prime degree is transitive and contains a Sylow $p$-subgroup $P$. Its normalizer in $A_p$ has order $pq$. For a proper normal subgroup $N$, the Frattini argument gives $G=N{N}_G(P)$; unless $G=C_p$, its $P$-normalizer is the full order-$pq$ normalizer, while $N_N(P)=P$. The Burnside transfer theorem gives a characteristic normal $p$-complement in $N$. This complement is normal in $G$ and cannot be transitive because its order is prime to $p$, so it is trivial. Thus $N=P$ and $|G|=pq$.

The theorem classifies finite primitive permutation groups by their minimal normal subgroups and the action of their socle. The modern types are affine (HA), almost simple (AS), simple diagonal (SD), compound diagonal (CD), product action (PA), holomorph simple (HS), holomorph compound (HC), and twisted wreath (TW). A coarser version describes primitive maximal subgroups of symmetric groups by affine, almost-simple, diagonal and full product-action overgroups. Intransitive and imprimitive maxima are respectively unequal-part subset stabilizers and block-system stabilizers. Primitivity alone is not maximality in the symmetric group; parity can put an otherwise natural candidate entirely inside the alternating group.

For a nonabelian simple group $T$, act on cosets of its full diagonal in $T^r$. A suitable factor-permuting extension is primitive and has point stabilizer intersecting the socle in the diagonal. The full normalizer has order $|T{|}^r|\mathrm{Out}(T)|r!$. For $T=A_5,r=2$, it acts on the sixty elements of $A_5$ by left/right multiplication, automorphisms and inversion, and has order $14400$. Odd conjugation in $S_5$ induces an odd permutation of these sixty points, while inversion is even; the full group is a primitive maximal subgroup of $S_{60}$.

An elementary abelian regular minimal normal subgroup acts as translations of a vector space. Its point stabilizer acts faithfully and irreducibly on that space; invariant subspaces would be blocks. The full affine normalizer is $AG{L}_d(p)$. In degree sixteen it lies inside $A_{16}$: nonzero translations swap eight pairs, and elementary linear transvections swap four pairs. Thus it is not a maximal subgroup of $S_{16}$.

The group of maps $x\mapsto Ax+b$ of a finite-dimensional vector space, with $A$ invertible. Identifying the underlying points with a regular elementary abelian group identifies its normalizer in the symmetric group with this semidirect product.

Conjugate the three-cycle throughout the group. Connected components of the support hypergraph form an invariant block system. Primitivity makes it connected, and connected triple supports generate an alternating group gives the conclusion.