A group G is• A set together with a binary map GxG®G sending the pair (p,q) of elements of G into pq and satisfying:• The associative law: (pq)r=p(qr)• The existence of an identity element: there exists an element e in G such that ep=pe=p for all p in G• The existence of a two sided inverse: for every p in G there is an element p-1 such that p-1p=pp-1 =e.• Group theory arose out of number theory, the theory of equations, crystallography, and geometry, but the first definition of an abstract group was given by Cayley in 1854.Arthur Cayley 1821 - 1895Examples: The special and general linear groups.Examples: The unitary and orthogonal groupsThe unitary group U(n) consists of all complex nÕn matrices which satisfy AA*=Iwhere I is the identity matrix. The subgroup SU(n) consist of thoseelements of U(n) which have determinant one. The orthogonal group O(n) consists of all real nÕn matrices which satisfy AAÖ= I.The subgroup SO(n) consists of those elements of O(n) with determinant one. The columns of an element of O(n) are unit vectors and any two distinct columns are orthogonal. Similarly for U(n).The group O(2).The group SO(3)A theorem of Euler asserts that if AÎ SO(3) and A ¹ I then Ais rotation about some axis. To prove this, it is enough to show that there is a non-zero vector v such that Av=v, because thenthe line through v is fixed by A, as is the plane perpendicular tov, and the restriction of A to that plane acts as an element of SO(2)and so is a rotation. To show that v exists we must show that A-I has a non-zero kernel, i.e. that det(A-I)=0. But det A=det AÖ=1so det(A-I) = det (AÖ-I) =(det A)(det (AÖ-I) = det[A(AÖ-I)] = det[AAÖ-A] =det(I-A) = det[(-I)(A-I) = det(-I)det(A-I) = -det(A-I).So det(A-I)=0 proving Euler’s theorem.The permutation groupsThe ax+b group.The ax+b group, continued.The Affine group A(n).Aff(n), continued.The Euclidean group E(n).This is the subgroup of Aff(n) where the A is restricted to beorthogonal. So an element of E(n) is of the form (A,v) whereAA†=I. The action of (A,v) on a vector x is given by (A,v)x = Ax+v.First apply the linear transformation A to x and then applythe translation through the vector v. In group language, (A,v)=(I,v)(A,0)The group E(2) is the group of “congruences” of Euclideanplane geometry.Let T denote the subgroup consisting of all translations, soT denotes the set of all elements of the form (I,v). Let H denote The subgroup consisting of all (A,0).Conjugation and normal subgroupsLet G be any group. If a Î G, the conjugation action by a is the map ofG into itself given by b ˚ aba-1 .We have a(bc)a-1 = (aba-1 )(aca-1),by the associative law. So conjugation by a is an automorphism of G.A subgroup N of G is called normal if every element of N is carried into N by conjugation by every element of G. In symbols n Î N implies ana-1 Î N for all a Î G.We claim that T is a normal subgroup of Aff(n) and of E(n). Indeed,(A,w)(I,v)(A,w) -1 =(A,w)(I,v)(A -1 ,-A-1w) =(A,Av+w)(A-1,-A-1w)= =(I, Av) Î N .The conjugation action action of an element (A,w) on (I,v) Î T sends(I,v) into (I,Av).Semi-direct productWe generalize the example of Aff(n) or E(n) as follows: Let N be agroup. For applications we will assume that N is commutative, andwrite the group composition law as addition (i.e. with a “+” sign). Let H be some other group (not necessarily commutative) and writeits group law as usual. Suppose that H “acts as automorphisms” of N.This means that any A Î H sends n Î N into an element An, andwe have A(n+m)= An+Am and A(Bn)=(AB)n for all A,B Î H and m,n Î N .Then we can construct a group whose elements are all pairs (A,n)with A Î H and n Î N, and the multiplication law is (A,m)(B,n) = (AB, An+m).This group is called the semi-direct product of H and N. We canIdentify N as the normal subgroup of this semi-direct product consistingof all elements of the form (I,n) where I is the identity element of H. Also H can be identified with the set of all elements of the form (A,0). Notice that in this identification HÇN consists only of the identity (I,0).Semi-direct products from an internal viewpoint.Conversely, suppose we start with a group G which contains anormal subgroup N, and also contains a subgroup H with the properties HÇN = identity element, and G =NH.The second equation means that every element of G can be written as a product mA with mÎ N and AÎ H. Then mAnB=m(AnA-1)AB.So if we define the action of H on N by A•n:= AnA-1 (the notationis a little confusing) we see that G is the semi-direct product of H! and N. Example: G = S3, N = C3 the cyclic group of order three (consisting)of those permutations preserving the cyclic order, and H = S{2,3} thepermutation group on the two elements 2 and