Section FrobeniusProperties.
Variables G H K : {group gT}.
Hypothesis frobG : [Frobenius G = K ><| H].

(* Specifications: *)
Lemma FrobeniusWker : [Frobenius G with kernel K].
Lemma Frobenius_partition : partition (gval K |: (H^# :^: K)) G.
Lemma Frobenius_cent1_ker : {in K^#, forall x, 'C_G[x] \subset K}.
Lemma Frobenius_reg_ker : semiregular K H.
Lemma Frobenius_reg_compl : semiregular H K.
Lemma ltn_odd_Frobenius_ker : odd #|G| -> #|H|.*2 < #|K|.
Lemma Frobenius_index_dvd_ker1 : #|G : K| %| #|K|.-1.
Lemma Frobenius_coprime : coprime #|K| #|H|.
Lemma Frobenius_index_coprime : coprime #|K| #|G : K|.
Lemma Frobenius_compl_Hall : Hall G H.
End FrobeniusProperties.
Lemma normedTI_J x A G L : normedTI (A :^ x) (G :^ x) (L :^ x) = normedTI A G L.
Lemma FrobeniusJcompl x G H :
[Frobenius G :^ x with complement H :^ x] = [Frobenius G with complement H].
Lemma FrobeniusJ x G K H :
[Frobenius G :^ x = K :^ x ><| H :^ x] = [Frobenius G = K ><| H].
Lemma FrobeniusJker x G K :
[Frobenius G :^ x with kernel K :^ x] = [Frobenius G with kernel K].
Lemma FrobeniusJgroup x G : [Frobenius G :^ x] = [Frobenius G].
Lemma Frobenius_ker_dvd_ker1 G K :
[Frobenius G with kernel K] -> #|G : K| %| #|K|.-1.
Lemma prime_FrobeniusP G K H :
K :!=: 1 -> prime #|H| ->
reflect (K ><| H = G /\ 'C_K(H) = 1) [Frobenius G = K ><| H].
Lemma Frobenius_subl G K K1 H :
K1 :!=: 1 -> K1 \subset K -> H \subset 'N(K1) -> [Frobenius G = K ><| H] ->
[Frobenius K1 <*> H = K1 ><| H].
Lemma Frobenius_subr G K H H1 :
H1 :!=: 1 -> H1 \subset H -> [Frobenius G = K ><| H] ->
[Frobenius K <*> H1 = K ><| H1].
Lemma Frobenius_kerP G K :
reflect [/\ K :!=: 1, K \proper G, K <| G
& {in K^#, forall x, 'C_G[x] \subset K}]
[Frobenius G with kernel K].
Lemma set_Frobenius_compl G K H :
K ><| H = G -> [Frobenius G with kernel K] -> [Frobenius G = K ><| H].
Lemma Frobenius_kerS G K G1 :
G1 \subset G -> K \proper G1 ->
[Frobenius G with kernel K] -> [Frobenius G1 with kernel K].
Lemma Frobenius_action_kernel_def G H K sT S to :
K ><| H = G -> @Frobenius_action _ G H sT S to ->
K :=: 1 :|: [set x in G | 'Fix_(S | to)[x] == set0].