Set of all one-to-one mappings of a non empty set onto itself

[itznogood]

Herstein in his book topics of algebra(sec 1.2, 2nd ed) defines A(S) to be set of all one-to-one mappings of S onto itself, S is a non empty set. Is he defining A(S) to be set of all bijections from S-->S or is he defining A(S) to be set of all injections from S--->S.

He uses the word one-to-one as well as "S 'onto' itself". Does the "onto" mean surjective here.
(if that be the case being injective and surjective makes the mapping bijective , but if onto doesn't mean surjective then A(S) is set of all injections)

I am confused because in next page , there is a problem where for S being a finite set he asks us to prove
"if σ is a one-to-one mapping of S onto itself then σ is onto".

So inference I am drawing is "mapping of S onto itself" doesn't mean necessarily that the mapping is onto/surjective as he is asking us to prove so in a special case. Am I right in my inference?

I wish Herstein used the terms "injective","surjective","bijective" instead of these confusing terms such as "onto" and "onto itself".I am unable to understand the groups chapter because of this,because Herstein makes frequent use of A(S) to explain groups concepts. So, is A(S) the set of all injections from S--->S or is it set of all bijections from S---> S ?

*This isn't exactly homework problem but is a doubt regarding explanation given in a textbook.As I am new here,I didn't know where to post so posted in homework section since the doubt is textbook style question. Please move this thread to the apt forum if it shouldn't be here*

 

[lanedance]

surjective = onto
injective = into
bijective = onto & into

I agree the "if σ is a one-to-one mapping of S onto itself then σ is onto" is a little confusing, maybe the best would be somtehing like"if σ is a one-to-one mappings of S to itself then σ is onto" us better

note it is not true in the infinite case

 

[itznogood]

Yup, that's what. "S to itself" , "S--->S" all these are ok.

And yes I am concerned about cases when S is infinite .

Coz for finite set S the mapping "S--->S" if it is injective will be surjective too and hence also bijective.

But for the infinite case this may not be so.Hence I want to make sure what exactly A(S) is.
is it set of all injective mappings or set of all bijective mappings of the type S--->S ?
Anyone well acquainted with Herstein book can help me out here.

I wouldn't have bothered in case it was just a one off problem. Trouble is he has explained a lot of group concepts using A(S) as a particular case of groups.for finite groups I am sure what A(S) is but not for the infinite cases. Note:- I think if A(S) is set of all injective mappings from S--->S , then set of all bijective mappings from S--->S will be a subset of A(S). As bijective is both into and onto.

 

[itznogood]

I think A(S) has to be set of all bijections from S--->S.

Because on the next page Herstein lists properties of A(S).

He says for any element σ in A(S) , there exists an element σ^{-1} in A(S).

I guess, an inverse mapping is possible only for bijective maps.

I hope I am right , still not 100% sure though.

 

[mathwonk]

i think he means A(S) = all bijections. and unfortunately he seems to have slipped in the problem which should be to prove a self map of a finite set is injective iff surjective.

i recommend mike artin's Algebra as a better book by the way.