Proposition 66. Let a finite group D act on S=∏_ i∈I T_i , where all T_i are isomorphic to a nonabelian simple group T , and suppose that its action on I is transitive. Fix i_0∈I , set A=D_ i_0 , identify T_ i_0 with T , and let α:A(T) be …

Use left conjugation for the given action and let π_0:S→T be the chosen coordinate projection. The map s f_s, f_s(d)=π_0(d·s), identifies S with F_α = f:D→T:f(ad)=α(a)(f(d)) for a∈A,d∈D . Indeed, values on representatives of A D are independent coordinates of S . Under this identification the action is (g·f)(d)=f(dg) . An extension β defines F_β = f:D→T:f(ud)=β(u)(f(d)) for u∈U,d∈D . This is a D -invariant subgroup of F_α , with independent values on the cosets in U D . Consequently F_β≅T^ [D:U] , and it projects onto every coordinate of F_α . Set S(U,β)= F_β in these coordinates. Conversely, let K≤ F_α be a D -invariant subdirect subgroup. Its diagonal-strip partition is invariant under the transitive action on A D . The block containing A has stabilizer U for some A≤U≤D . Projection at 1 identifies the diagonal factor on that block with T . The induced action of U on this factor gives a homomorphism β:U(T) extending α . For f∈K and u∈U one has f(u)=β(u)(f(1)) . Applying this identity to d·f gives f(ud)=β(u)(f(d)). Thus K≤ F_β . Both groups have one freely varying diagonal factor for each coset of U , so they are equal. Containment of two subdirect groups reverses refinement of their strip partitions. Hence S(U,β)≤S(V,γ) forces V≤U . For v∈V , varying f(1) over T in the two defining equations then gives β(v)=γ(v) . The converse follows immediately from those equations. Finally, every subgroup containing a subdirect subgroup is subdirect, so all members of the stated interval have already been included.