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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]>]]]]
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