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Appendix: Elements of Group Theory

1. Semidirect Product and Its Properties

Definition A.1.1. Let G1 and G2 be the groups and f: G1 ® Aut(G2) the homomorphism between these two groups, where Aut(G2) is the set of all automorphisms of the group G2. The semidirect product of G1 and G2 with the operation f is

G1×fG2 = {(g1,g2) | g1Î G1, g2 Î G2 },
(g1,g2)(g1¢,g2¢) = (g1f(g2)(g1¢),g2g2¢).

The identity is (e1,e2), where e1 is the identity element from G1, and e2 the identity from G2. The inverse element of (g1,g2) is (f(g2)-1(g1-1),g2-1). From

[(g1,g2)(g1¢,g2¢)] (g1¢¢,g2¢¢)
= (g1f(g2)(g1¢),g2g2¢)(g1¢¢,g2¢¢)
= (g1f(g2)(g1¢)f(g2g2¢)(g1¢¢),g2g2¢g2¢¢)  =f (g1f(g2)(g1¢f(g2¢)(g1¢¢)),g2g2¢g2¢¢)
= (g1,g2)(g1¢f(g2¢)(g1¢¢),g2¢g2¢¢)
= (g1,g2)[(g1¢,g2¢)(g1¢¢,g2¢¢)] ,

we conclude that it is associative. So, the semidirect product of two groups is a group.

Theorem A.1.1. Let T and O be the subgroups of G. If T is a normal subgroup of G, G=TO, TÇO = {e} (where e is the identity from G), then G is isomorphic to the semidirect product of T and O, G @ T×fO, where f: O® Aut(T), so that f(y)(x) = yx-1, x Î T, y Î O.

Proof: Let y: T×fO® G that y(x,y) = xy .

y[(x,y)(x¢,y¢)] = (xf(y)(x¢),yy¢)
= y(xyx¢y-1,yy¢) = xyx¢y-1yy¢
= xyx¢y¢ = y(x,y) y(x¢,y¢),

so y is a homomorphism. Since G = TO, y is a surjection. From y(x,y) = y(x¢,y¢) Þ xy = x¢y¢Þ x¢-1x= y¢y-1, as TÇO = {e}, it follows x = x¢ and  y = y¢. So, y is the injection. Because y is homomorphic surjective and injective function, we conclude that y is an isomorphism.

2. Orbit

Definition A.2.1. Let G be the group of transformations of some set X. The orbit of an element x Î X under the group G is OrbG(x) = {g(x) | g Î G}.

3. Discrete Groups

Definition A.3.1. The group of transformations G of the set X is discrete if "x Î X, OrbG(x) is a discrete set, that is ($d > 0)("g1,g2 Î G)(d < |g1(x)-g2(x)|).

3. Group Realization

Let G1 and G2 be the groups and let their semidirect product be defined as in Definition A.1.1.

Definition A.4.1. The realization of the element g1 Î G1 (g2 Î G2, respectively) in the semidirect product of the groups G1 and G2 is the set of all ordered pairs (g1,x) ((x,g2), respectively) from G1×fG2.