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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)%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]>]]]] | |