cyclic_pgroup_Aut_structure
     : forall (gT : finGroupType) (p : nat) (G : {group gT}),
       p.-group G ->
       cyclic G ->
       G != 1 ->
       let q := #|G| in
       let n := (logn p q).-1 in
       let A := Aut G in
       let P := 'O_p(A) in
       let F := 'O_p^'(A) in
       exists m : {perm gT} -> 'Z_q,
         [/\ [/\ {in A & G, forall (a : {perm gT}) (x : gT), x ^+ m a = a x},
                 m 1 = 1%R /\ {in A &, {morph m : a b / a * b >-> (a * b)%R}},
                 {in A &, injective m} /\ [seq m x | x in A] =i GRing.unit,
                 forall k : nat,
                 {in A, {morph m : a / a ^+ k >-> (a ^+ k)%R}}
               & {in A, {morph m : a / a^-1 >-> (a^-1)%R}}],
             [/\ abelian A, cyclic F, #|F| = p.-1
               & [faithful F, on 'Ohm_1(G) | [Aut G]]]
           & if n == 0
             then A = F
             else
              exists t : perm_for_finType gT,
                [/\ t \in A, #[t] = 2, m t = (-1)%R
                  & if odd p
                    then
                     [/\ cyclic A /\ cyclic P,
                         exists s : perm_for_finType gT,
                           [/\ s \in A, #[s] = (p ^ n)%N, m s = (p.+1%:R)%R
                             & P = <[s]>]
                       & exists s0 : perm_for_finType gT,
                           [/\ s0 \in A, #[s0] = p, m s0 = ((p ^ n).+1%:R)%R
                             & 'Ohm_1(P) = <[s0]>]]
                    else
                     if n == 1
                     then A = <[t]>
                     else
                      exists s : perm_for_finType gT,
                        [/\ s \in A, #[s] = (2 ^ n.-1)%N, 
                           m s = 5%R, <[s]> \x <[t]> = A
                          & exists s0 : perm_for_finType gT,
                              [/\ s0 \in A, #[s0] = 2,
                                  m s0 = ((2 ^ n).+1%:R)%R,
                                  m (s0 * t) = ((2 ^ n).-1%:R)%R
                                & 'Ohm_1(<[s]>) = <[s0]>]]]]