Proving Surjectivity of Mapping A_g:G ---> G for Automorphism

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In summary, the conversation discussed a mapping A_g: G ---> G defined as A_g(x) = g^-1(x)g for all x in G, and the task of showing its surjectivity as part of proving it is an automorphism. The definition of surjectivity was mentioned, as well as the criteria for a mapping to be surjective. An attempt was made to show surjectivity using the explicit pre-image of an element x in G. The person asking for feedback or hints was satisfied with the explanation given.
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RVP91
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I have a mapping A_g:G ---> G defined by
A_g(x) = g^-1(x)g (for all x in G)

and as part of showing it is an automorphism i need show it is surjective.

I'm not entirely sure how to do this but have made an attempt and would appreciate and feedback or hints to what I actually need to show. I know the definition of surjectivity and also that a mapping is surjective iff Im(of mapping) = G

My attempt:
It is surjective since if x is in G, then gxg^-1 is in G and then A_g(gxg^-1) = (g^-1)gx(g^-1)g = idxid = x.

Thanks in advance
 
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  • #2
Yep, surjective just means that every element has a pre-image, and you have shown that by writing down the pre-image explicitly.
 
  • #3
thanks
 

1. What does it mean for a function to be surjective?

Surjectivity is a property of a function where every element in the codomain has at least one preimage in the domain. In other words, every element in the output has at least one input that maps to it.

2. How can I show that a function is surjective?

To show that a function is surjective, you need to prove that every element in the codomain has at least one preimage in the domain. This can be done by choosing an arbitrary element in the codomain and finding its preimage in the domain.

3. Can a function be both surjective and injective?

Yes, a function can be both surjective and injective. This is known as a bijective function, where every element in the codomain has exactly one preimage in the domain.

4. If a function is surjective, does it mean it is also onto?

Yes, surjective and onto are two terms that mean the same thing in the context of functions. A function is said to be surjective or onto if every element in the codomain has at least one preimage in the domain.

5. Are there any visual representations of surjective functions?

Yes, there are visual representations of surjective functions. One common representation is the arrow diagram, where arrows connect elements in the domain to their corresponding elements in the codomain. In a surjective function, every element in the codomain will have at least one arrow pointing to it.

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