80a:20004 20B05 (20B15)
Goldschmidt, David M.; Scott, Leonard L.
A problem of W. A. Manning on primitive permutation groups.
Math. Z. 161 (1978), no. 2, 97--100.

Let G be a transitive permutation group on a finite set Omega . For each orbit Delta of the stabilizer G{sub} alpha of a point alpha {in} Omega , Delta {sup*}=(alpha {sup}g{vert} alpha {sup}(g - 1){in} Delta , g {in}G) is also an orbit of G{sub} alpha , and it is said to be paired with Delta . Let G{sub}(alpha , Delta ) denote the pointwise stabilizer of Delta in G{sub} alpha . In 1927, W. A. Manning posed the following problem: (*) Suppose that G is primitive on Omega . Is it true that G{sub}(alpha , Delta )=G{sub}(alpha , Delta {sup*}) for all orbits Delta of G{sub} alpha ? Positive results of a related nature (such as relations between G {sub}(alpha , Delta ) and G{sub}(alpha , Delta {sup*})) have been obtained by many authors. However, the present authors note that the question (*) is equivalent to (**) Suppose that N{sub}G(V) is a maximal subgroup of a finite group G for some subgroup V{lhkeq}G. Is the relation "A normalizes B" necessarily symmetric on the G-conjugates of V (namely, if V{sup}x normalizes V{sup}y, then does V{sup}y normalize V{sup}x)? They point out the existence of a large family of counterexamples to (**), in which G is a symmetric or alternating group and V is a regular elementary abelian p-subgroup.