4. For any sets A, B and any function f: A → B, we define the preimage of a subsetSCB to be the set {a € A | ƒ(a) ≤ S}. We use the notation f-¹(S) to denote thepreimage of S, even though ƒ may not have an inverse function.Show that if a: Gthen keraa ¹(S) ◄ G.H is a group homomorphism, and SH is a subgroup,a ¹(S) C G and a ¹(S) is a subgroup. Moreover, if S ◄ H, then

Answered: Image /qna-images/answer/4e016aa6-17e7-46d7-b323-dfc14c5d25c5.jpg

4. For any sets A, B and any function f: A → B, we define the preimage of a subset SCB to be the set {a e A | f(a) e S}. We use the notation f-¹(S) to denote the preimage of S, even though f may not have an inverse function. Show that if a: G then ker a a ¹(S) ◄ G. H is a group homomorphism, and SC H is a subgroup, a¹(S) C G and a-¹(S) is a subgroup. Moreover, if S ◄ H, then

4. For any sets A, B and any function f: A → B, we define the preimage of a subset SCB to be the set {a e A | f(a) e S}. We use the notation f-¹(S) to denote the preimage of S, even though f may not have an inverse function. Show that if a: G then ker a a ¹(S) ◄ G. H is a group homomorphism, and SC H is a subgroup, a¹(S) C G and a-¹(S) is a subgroup. Moreover, if S ◄ H, then

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.3: Subgroups

Problem 23E: 23. Let be the equivalence relation on defined by if and only if there exists an element in ...

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