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A group scheme that is finite and flat over the ground scheme. If  $  G $
is a finite group scheme over a scheme  $  ( G, {\mathcal O} _ {S} ) $, 
then  $  G =  \mathop{\rm Spec}  {\mathcal A} $, 
where  $  {\mathcal A} $
is a finite flat quasi-coherent sheaf of algebras over  $  {\mathcal O} _ {S} $. 
From now on it is assumed that  $  S $
is locally Noetherian. In this case  $  {\mathcal A} $
is locally free. If  $  S $
is connected, then the rank of  $  {\mathcal A} \times _ { {\mathcal O} _ {s}  } k ( s) $
over the field of residues  $  k ( s) $
at a point  $  s \in S $
is independent of  $  s $
and is called the rank of the finite group scheme. Let  $  n _ {G} :  G \rightarrow G $
be the morphism of  $  S $-
schemes mapping an element  $  s \in G ( T) $
into  $  x  ^ {n} \in G ( T) $, 
where  $  T $
is an arbitrary  $  S $-
scheme. The morphism  $  n _ {G} $
is null if the rank of  $  G $
divides  $  n $
and if  $  S $
is a reduced scheme or if  $  G $
is a commutative finite group scheme (see [[Commutative group scheme|Commutative group scheme]]). Every finite group scheme of rank  $  p $, 
where  $  p $
is a prime number, is commutative [[#References|[2]]].

If  $  G _ {1} $
is a subgroup of a finite group scheme  $  G $, 
then one can form the finite group scheme  $  G/G _ {1} $, 
and the rank of  $  G $
is the product of the ranks of  $  G _ {1} $
and  $  G/G _ {1} $.

===Examples.===

1) Let  $  G = G _ {mS} $
be a multiplicative group scheme (or Abelian scheme  $  {\mathcal A} $
over  $  S $); 
then  $   \mathop{\rm Ker}  n _ {G} $
is a finite group scheme of rank  $  | n | $(
or  $  | n | ^ {2   \mathop{\rm dim}  {\mathcal A} } $).

2) Let  $  S $
be a scheme over the prime field  $  \mathbf F _ {p} $
and let  $  F:  G _ {aS} \rightarrow G _ {aS} $
be the Frobenius homomorphism of the additive group scheme  $  G _ {aS} $. 
Then  $   \mathop{\rm Ker}  F $
is a finite group scheme of rank  $  p $.

3) For every abstract finite group scheme  $  \Gamma $
of order  $  n $
the constant group scheme  $  \Gamma _ {S} $
is a finite group scheme of rank  $  n $.

The classification of finite group schemes over arbitrary ground schemes has been achieved in the case where the rank of  $  G $
is a prime number (cf. [[#References|[2]]]). The case where  $  G $
is a commutative finite group scheme and  $  S $
is the spectrum of a field of characteristic  $  p $
is well known (see [[#References|[1]]], [[#References|[3]]], [[#References|[7]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.I. Manin,   "The theory of commutative formal groups over fields of finite characteristic"  ''Russian Math. Surveys'' , '''18'''  (1963)  pp. 1–80  ''Uspekhi Mat. Nauk'' , '''18''' :  6  (1963)  pp. 3–90</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Tate,   F. Oort,   "Group schemes of prime order"  ''Ann. Sci. Ecole Norm. Sup.'' , '''3'''  (1970)  pp. 1–21</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Demazure,   P. Gabriel,   "Groupes algébriques" , '''1''' , Masson  (1970)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  F. Oort,   "Commutative group schemes" , ''Lect. notes in math.'' , '''15''' , Springer  (1966)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  S. Shatz,   "Cohomology of Artinian group schemes over local fields"  ''Ann. of Math.'' , '''79'''  (1964)  pp. 411–449</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  B. Mazur,   "Notes on étale cohomology of number fields"  ''Ann. Sci. Ecole Norm. Sup.'' , '''6'''  (1973)  pp. 521–556</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  H. Kraft,   "Kommutative algebraische Gruppen und Ringe" , Springer  (1975)</TD></TR></table>

====Comments====
For some spectacular applications of the results in [[#References|[a1]]] see [[#References|[a2]]], [[#References|[a3]]].

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Raynaud,   "Schémas en groupes de type $(p,\ldots,p)$"  ''Bull. Soc. Math. France'' , '''102'''  (1974)  pp. 241–280</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  J.-M. Fontaine,   "Il n'y a pas de variété abélienne sur $\ZZ$"  ''Invent. Math.'' , '''81'''  (1985)  pp. 515–538</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Faltings,   "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern"  ''Invent. Math.'' , '''73'''  (1983)  pp. 349–366  (Erratum: Invent. Math '''75''' (1984), 381)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G. Cornell (ed.)  J. Silverman (ed.) , ''Arithmetic geometry'' , Springer  (1986)</TD></TR>
</table>