$p$-divisible groups
A $p$-divisible group is a tower of affine group schemes $$G_{1}\subset G_{2}\subset G_{3}\subset G_{4}\subset \cdot\cdot\cdot$$ such that some extra conditions are satisfied.
1. Definition
A $p$-divisible group (or Barsotti-Tate group) of height $h$ over a Noetherian ring $k$ is a sequence of affine groups schemes $G_1, G_2, G_3, . . .$ over $k$, together with a morphism $i_n : G_n \to G_{n+1}$ for each $n \geq 1$, satisfying the following for each $n \geq 1$:
(i) $G_n$ is finite flat of order $p^{nh}$;
(ii) the morphism $i_ n$ is injective and has image $G_{n+1}[p^{n}]$.
In other words, up to identifying $G_n$ with its image in $G_{n+1}$, a $p$-divisible group is a tower of affine group schemes $$G_{1}\subset G_{2}\subset G_{3}\subset G_{4}\subset \cdot\cdot\cdot$$ such that $G_{n} = G_{n+1}[p^{n}]$ for all $n \geq 1$, together with the condition that $G_{n}$ is finite flat of order $p^{nh}$.