1. Section 13 Homomorphisms Definition A map  of a group G into a group G’ is a homomorphism if the homomophism property  (ab) =  (a)  (b) Holds for all a, b  G. Note: The above equation gives a relation between the two group structures G and G’. Example: For any groups G and G’, there is always at least one homomorphism:  : G  G’ defined by  (g)=e’ for all g  G, where e’ is the identity in G’. We call it the trivial homomorphism.

2. Examples Example Let r  Z and let  r : Z  Z be defined by  r (n)=rn for all n  Z. Is  r a homomorphism? Solution: For all m, n  Z, we have  r (m + n) = r(m + n) = rm + rn =  r (m)+  r (n). So  r is a homomorphism. Example: Let  : Z 2  Z 4  Z 2 be defined by  (x, y)=x for all x  Z 2, y  Z 4. Is  a homomorphism? Solution: we can check that for all (x 1, y 1 ), (x 2, y 2 )  Z 2  Z 4,  ((x 1, y 1 )+(x 2, y 2 ) )= x 1 + x 2 =  (x 1, y 1 )+  (x 2, y 2 ). So  is a homomorphism.

3. Composition of group homomorphisms In fact, composition of group homomorphisms is again a group homomorphism. That is, if  : G  G’ and  : G’  G’’ are both group homomorphisms then their composition (   ): G  G’’, where (   )(g)=  (  (g)) for g  G. is also a homomorphism. Proof: Exercise 49.

4. Properties of Homomorphisms Definition Let  be a mapping of a set X into a set Y, and let A  X and B  Y. The image  [A] of A in Y under  is {  (a) | a  A}. The set  [X] is the range of . The inverse image of  -1 [B] of B in X is {x  X |  (x)  B }.

5. Theorem Let  be a homomorphism of a group G into a group G’. 1.If e is the identity element in G, then  (e) is the identity element e’ in G’. 2.If a  G, then  (a -1 )=  (a) -1. 3.If H is a subgroup of G, then  [H] is a subgroup of G’. 4.If K’ is a subgroup of G’, then  -1 [K’] is a subgroup of G. Proof of the statement 3: Let H be a subgroup of G, and let  (a) and  (b) be any two elements in  [H]. Then  (a)  (b)=  (ab), so we see that  (a)  (b)   [H]; thus  [H] is closed under the operation of G’. The fact that e’=  (e) and  (a -1 )=  (a) -1 completes the proof that  [H] is a subgroup of G’.

6. Kernel Collapsing Definition Let  : G  G’ be a homomorphism of groups. The subgroup  -1 [{e’}]={x  G |  (x)=e’} is the kernel of , denoted by Ker(  ). Let H= Ker(  ) for a homomorphism . We think of  as “collapsing” H down onto e’. G’ G a’  (b) e’  (x) y’  -1 [{a’}] bH H Hx  -1 [{y’}] b e x

7. Theorem Let  : G  G’ be a group homomorphism, and let H=Ker(  ). Let a  G. Then the set  -1 [{  (a)}]={ x  G |  (x)=  (a)} is the left coset aH of H, and is also the right coset Ha of H. Consequently, the two partitions of G into left cosets and into right cosets of H are the same. Corollary A group homomorphism  : G  G’ is a one-to-one map if and only if Ker(  )={e}. Proof. Exercise.

8. Normal Subgroup Definition A subgroup H of a group G is normal if its left and right cosets coincide, that is if gH = Hg for all g  G. Note that all subgroups of abelian groups are normal. Corollary If  : G  G’ is a group homomorphism, then Ker(  ) is a normal subgroup of G.