= Maximal symmetric-group subgroups containing a three-cycle
{title2=$A_n,\quad S_k\times S_{n-k},\quad S_a\wr S_b$}
Up to <conjugation>, the proper <maximal subgroups> of $S_n$ containing a <three-cycle> are $A_n$; the unequal-part subset stabilizers $S_k\times S_{n-k}$ for $1\leq k<n/2$ with $n-k\geq3$; and the full block stabilizers $S_a\wr S_b$ for $n=ab$, $a\geq3$, $b\geq2$. The <primitive three-cycle criterion> treats the primitive case. A nontrivial <block system> gives the full <permutation wreath product> in the imprimitive case. These block stabilizers are maximal because any extra <permutation> produces a transposition connecting two different blocks; conjugating by the block stabilizer then gives every cross-block transposition, hence the <symmetric group>. Unequal-part subset stabilizers are maximal by the same connected-transposition argument. Equal-part subset stabilizers are not maximal: they lie in the larger stabilizer that can interchange the two parts.
Back to article page
