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

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].