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§2. Group Actions on Sets and Groups 147

For s ∈ G we denote the action of s on x ∈ E by sx. Then the group homomor-
phism condition becomes
1x = x and s(tx) = (st)x for all s, t ∈ G, x ∈ E.
For a pointed set we also require sx 0 = x 0 , where x 0 is the base point. When E is an
additive group, we have s(x + y) = sx + sy,and when E is a multiplicative group,
s
we write frequently x instead of sx and the automorphism condition has the form
s s
s (xy) = x y.
A right G action on E is an antihomomorphism G → Aut(E), and the action of
s
s ∈ G on x ∈ E is written xs or x . The antihomomorphism condition is x1 = x
and (xs)t = x(st) so that sx = xs −1 is a left action associated with the right action.
Let G be a group. We can speak of left and right G-sets, pointed G-sets, and
G-groups, i.e., the corresponding object a set, a pointed set, or group together with
the corresponding G action.
(2.2) Definition. A morphism f : E → E of G-objects is a morphism of objects

E → E together with the G-equivariance property f (sx) = sf (x) (or f (xs) =
f (x)s) for all s ∈ G, x ∈ E.
(2.3) Example. Let D be a G-subgroup of a G-group E,so sD = D for all s ∈
G. Then D has by restriction a G-group structure and the inclusion D → E is a
morphism of G-groups. Now form the quotient pointed sets D\E of all right D-
cosets Dx and E/D of all left D-cosets xD, where x ∈ E. The base point is the
identity coset D.The G action on E induces a G action on the quotient pointed sets
D\E and E/D such that the projections E → D\E and E → E/D are morphisms
of pointed G-sets. When D is a normal subgroup of E, then E/D = D\E is a
G-group, and E → E/D is a morphism of G-groups. We write
1 → D → E → D\E → 1 and 1 → D → E → E/D → 1
as short exact sequences of pointed sets, where D → E is injective and im(D → E)
is the kernel of the surjection E → D\E or E → E/D.
G
(2.4) Definition. Let E be a G-object. The subobject of fixed elements E consists
of all x ∈ E with sx = x (or xs = x) for all s ∈ G.
G
If E is a G-set (resp. pointed G-set, G-group), then E is a subset (resp. pointed
0
subset, subgroup) of E. We denote E G also by H (G, E) meaning the zeroth co-
homology object of G with values in E. With this notation we anticipate the first
1
cohomology set H (G, E), where E is a G-group.

(2.5) Examples. Let k /k be a Galois extension with Galois group G =Gal(k /k).
0
G

Then the additive group k is a G-group and (k ) = H (G, k ) = k. The matrix

groups GL n (k ) and SL n (k ) are also G-groups, and the subgroups fixed by the action
G
G

are GL n (k ) = GL n (k) and SL n (k ) = SL n (k). Also det: GL n (k ) → GL 1 (k ) is
a morphism of G-groups. Finally, if E is an elliptic curve defined over k, then E(k )

G
is a G-group with E(k ) = E(k). This can be seen directly with affine coordinates
G
or from the fact that P n (k ) = P n (k) which is also checked by looking on each
affine piece defined by y j = 0.