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Prove that if G, H, and K are groups, and 8 : G -;. Hand : H -;. K are isomorphisms,then

Chapter 18, Problem 18.9

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QUESTION:

Prove that if G, H, and K are groups, and 8 : G -;. Hand : H -;. K are isomorphisms,then 08 : G -;. K is an isomorphism. (Use multiplication for the group operations.)

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QUESTION:

Prove that if G, H, and K are groups, and 8 : G -;. Hand : H -;. K are isomorphisms,then 08 : G -;. K is an isomorphism. (Use multiplication for the group operations.)

ANSWER:

Step 1 of 2

Consider the groups  and  are isomorphisms.

To prove that  is an isomorphism, we have to prove the following conditions,

     1.  is one-one

     2.  is onto

     3.  is homomorphism

As are isomorphisms, they are bisections, and hence their composition

 is also bijective. Therefore the conditions (1),(2), are satisfied for  

 

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