Definition 19: Let H; be a normal subgroup of the group G;.G/H; O is called quotient group where the operation e is defined on G/h by Hg oHg2' H(g *g2) o If G is a finite group, then G/H is aiso a finite group, and G/HFG H
Definition 19: Let [H;*] be a normal subgroup of the group [G;*]. [G/H;] is called quotient group, where the operation is defined on G/H by Hg1Hg2 = H(g1*g2 ). If G is a finite group, then G/H is also a finite group, and |G/H|=|G|/|H|
6.5 The fundamental theorem of homomorphism for groups 46.5.1.Homomorphism kernel and homomorphism image t Lemma 4: Let G; *I and IG; l be groups, and op be a homomorphism function from G to G. Then pleg) is identity element of IG;. that x=p(aP(GEG. Then 3 aEG such ◆ Proof:Letx
6.5 The fundamental theorem of homomorphism for groups 6.5.1.Homomorphism kernel and homomorphism image Lemma 4: Let [G;*] and [G';•] be groups, and be a homomorphism function from G to G'. Then (eG) is identity element of [G';•]. Proof: Let x(G)G'. Then aG such that x=(a)
. Example: [R-103, * and K-1, 13; *] are groups. 1x>0 p(x) Kerq={x|x>0,x∈R} -1x<0
Example: [R-{0};*] and [{-1,1};*] are groups. − = 1 0 1 0 ( ) x x x Ker ={x | x 0, xR}
O Definition 20: Let o be a homomorphism function from group G with identity element e to group G with identity element e. XEG(x) e is called the kernel of homomorphism function (p We denoted by Kero(k(p),or k)
Definition 20: Let be a homomorphism function from group G with identity element e to group G' with identity element e’. {xG| (x)= e'} is called the kernel of homomorphism function . We denoted by Ker( K(),or K)
Theorem 6.23: Let o be a homomorphism function from group G to group G. Then following results hold. ◆(1)Kerφ;] is a normal subgroup of[G; . (2)p is one-to-one iff K=(ed) ◆(3)|p(G;] is a subgroup of g';°l proof:(1)i Kero is a subgroup of g ◆ For va,b∈kerq,a2b∈?kerq, ◆ie.g(a2b)=?ec ◆ Inverse element: For va∈Kerq,al∈?Kerp ◆i) For GeC,2a∈Kerq,g1*a*g∈?Kerq
Theorem 6.23：Let be a homomorphism function from group G to group G'. Then following results hold. (1)[Ker;*] is a normal subgroup of [G;*]. (2) is one-to-one iff K={eG} (3)[(G); •] is a subgroup of [G';•]. proof:(1)i) Ker is a subgroup of G For a,bKer, a*b?Ker, i.e.(a*b)=?eG‘ Inverse element: For aKer, a -1?Ker ii)For gG,aKer, g-1*a*g?Ker