1 / 8

Section 13 Homomorphisms

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’.

marsha
Download Presentation

Section 13 Homomorphisms

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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 : Z2  Z4  Z2 be defined by (x, y)=x for all x  Z2, y  Z4. Is  a homomorphism? Solution: we can check that for all (x1, y1), (x2, y2)  Z2  Z4, ((x1, y1)+(x2, y2) )= x1+ x2= (x1, y1)+ (x2, y2). 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 Theorem Let  be a homomorphism of a group G into a group G’. • If e is the identity element in G, then (e) is the identity element e’ in G’. • If a  G, then (a-1)= (a)-1. • If H is a subgroup of G, then [H] is a subgroup of G’. • 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’. -1[{a’}] bH H Hx -1[{y’}] G b e x G’ a’ (b) e’ (x) y’

  7. Theorem 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.

More Related