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