Definition A.1
[Tat67], 1.1 A finite group scheme of rank $m$ over a base scheme $S$ is a finite morphism $G\to S$ such that $G$ is locally free of rank $m$ such that there is multiplication $m: G\times_S G\to G$, inverse $i:G\to G$, and $e: S\to G$ over $S$ determining a group structure.
Definition A.1
[Tat67], 1.3 Assume everything to be commutative, a short exact sequence of finite group schemes is a sequence of group scheme morphism \[ 0\to G’\xrightarrow{i} G\xrightarrow{p} G”\to 0 \] such that this is exact in the category of fppf sheaves of abelian group over schemes over $S$.
This is equivalent to closed immersion $i:G’\to G$ such that $G’=G\times_{G”}S$, and $p: G\to G”$ is faithfully flat.
Definition A.1
A $p$-divisible group of height $h$ over $S$ is a inductive system $(G_n, i_n)_{n\in \mathbb{N}}$ such that $G_n$ is a finite group of rank $p^{nh}$ over $S$, and closed immersions $i_n: G_n\to G_{n+1}$ such that \[ 0\to G_n \xrightarrow{i_n} G_{n+1}\xrightarrow{p^n}G_{n+1}\to 0 \] are exact.
A homomorphism of $p$-divisible group is a morphism of inductive system.
Examples
Let $G_n$ be ordinary abelian groups, then we have $G_1=(\Z/p\Z)^h$, and by induction, \[ G_n=(\Z/p^n\Z)^h, G=\colim_n G_n =(\mathbb{Q}_{p}/\Z_p)^h \]
Let $F$ be a formal group of height $h$, $F[p^n]$ of $p^n$-torsion are finite group schemes, and determines a $p$-divisible group $F[p^\infty]$. If $F$ has height $\geq h$, then $F[p^\infty]$ woud have at least height $h$.
For every finite group scheme $G$ over $S$, there is an short exact sequence [Tat67], 1.4 \[ 0\to G^0\to G\to G^{ét}\to 0 \] Where $G^0$ is connected, and $G^{ét}$ is étale over $S$. If $G$ is $p$-divisible, then $i_n:G_n^0\to G_{n+1}^0$, then there is an exact sequence of $p$-divisible group.