1.1. FUNDAMENTAL NOTIONS 3
between this category and the ordinary category of groups, we shall call the objects
in the latter category "abstract" groups.
PROPOSITION
1.1.2. Let K be an algebraic group. Then the following condi-
tions hold:
(a) If H is a closed subgroup of K, then H is an algebraic group whose poly-
nomial functions are the restrictions to H of those of K. Moreover, the
inclusion map H K is a morphism;
(b) // in addition H \ K, then K/H may be given the structure of algebraic
group {i.e., identified with a closed subgroup o/GLm(jP) for some m) in such
a way that the projection n : K K/H is a morphism, and every mor-
phism K K whose kernel contains H factors uniquely as the composite
of morphisms n : K K/H K;
(c) For any morphism K H of algebraic groups, its kernel and image are
closed subgroups of K and H, respectively;
(d) If Ki and K2 are algebraic groups, then their direct product K\ x K2 can
be given the structure of algebraic group in such a way that the canonical
injections Ki if i x K2 and projections K\xK2 Ki(i = 1,2) are mor-
phisms. Furthermore, K\X K2 is then a product in the category of algebraic
groups, and F[K\ x K2] = i^i^i] ®^F[K2]. If Ki is a closed subgroup of
GL(Vi), i = 1,2, then the obvious embedding K1 x K2 GL{V\ ® V2) is
an isomorphism of algebraic groups between K\ x K2 and its image;
(e) If H is a closed subgroup ofK, then N-j^(H) and C^(H) are closed subgroups
of K; and
(f) // K is an algebraic group and X is a finite group of automorphisms of K
(as algebraic group) then the semidirect product S of K by X may be given
the structure of algebraic group in such a way that the normal subgroup of
S corresponding to K is a closed subgroup and the identification of it with
K is an isomorphism of algebraic groups.
Next we consider the idea of connectedness.
DEFINITION
1.1.3. Let K be an algebraic group. Then K is the connected
component of K (in the Zariski topology) containing the identity element.
PROPOSITION
1.1.4. Let K be an algebraic group. Then the following condi-
tions hold:
(a) K is the unique smallest closed subgroup of K of finite index;
(b) K is connected, and K = K if and only if K is connected;
(c) If(j): K H is a morphism of algebraic groups, then /(K ) = /(K)° H ;
(d) If H K is a closed subgroup, then H K ;
(e) The subgroup of K generated by any family of closed connected subgroups is
again closed and connected. Moreover, if H and K are closed subgroups of
K such that H is connected, then the commutator subgroup [H, K] is closed
and connected; and
(f) K is connected if and only if F[K] is an integral domain.
The last result underlies the definition of dimension.
DEFINITION
1.1.5. Let K be an algebraic group. If if is connected, then F(K)
is the quotient field of F[if], and the dimension dim(K) is the transcendence
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