Nessary and sufficient condition for homomorphism to be isomorphism.

In summary, a homomorphism is a function that preserves the structure of a given algebraic system, while an isomorphism is a special type of homomorphism that is both injective and surjective. A necessary condition for an isomorphism is that it must be both injective and surjective, while a sufficient condition is that it must also have a two-sided inverse. Determining if a homomorphism is an isomorphism is important for understanding relationships between algebraic systems and solving problems and proving theorems in mathematics.
  • #1
AAQIB IQBAL
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The necessary and sufficient condition for homomorphisim f of a group G into a group G' with kernel K to be isomorphism of G into G' is that k={e}
... THOUGH I AM ABLE TO PROVE THAT f IS ONE-ONE AND f IS HOMOMORPHISM (in converse part) BUT CAN'T GET ANY IDEA TO PROVE THAT f IS ONTO.
PLEASE HELP ME IN THIS REGARD
 
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  • #2
That Ker(f)={e} is a necessary and sufficient condition for f to be injective.
You won't be able to prove that f is an isomorphism, because it is false in general.
 

Related to Nessary and sufficient condition for homomorphism to be isomorphism.

What is a homomorphism?

A homomorphism is a mathematical function that preserves the structure of a given algebraic system. In simpler terms, it is a function that maps elements from one algebraic structure to another while preserving the operations and relations between them.

What is an isomorphism?

An isomorphism is a special type of homomorphism that is bijective, meaning it is both injective (one-to-one) and surjective (onto). This means that an isomorphism not only preserves the structure of the algebraic system, but also the individual elements and their relationships.

What is a necessary condition for a homomorphism to be an isomorphism?

A necessary condition for a homomorphism to be an isomorphism is that it must be both injective and surjective. In other words, every element in the range of the homomorphism must have a unique preimage in the domain, and every element in the domain must have at least one corresponding element in the range.

What is a sufficient condition for a homomorphism to be an isomorphism?

A sufficient condition for a homomorphism to be an isomorphism is that it must be both injective and surjective, as well as have a two-sided inverse. This means that there must exist another homomorphism in the opposite direction that reverses the mapping and also preserves the structure of the algebraic system.

Why is determining if a homomorphism is an isomorphism important?

Determining if a homomorphism is an isomorphism is important because it allows us to understand the relationship between different algebraic systems. Isomorphisms can help us identify underlying similarities and differences between structures, and can also be used to solve problems and prove theorems in various fields of mathematics.

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