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In Lang's algebra I see the following definition:

"Let $G$ be a finite cyclic group of order $n$, generated by an element $\sigma$. Assume that $G$ operates on a an abelian group $A$ and let $f:A\to A$ be the endomorphism of $A$ given by $f(x)=\sigma x - x$ ..."

Why is $f$ an endomorphism? I thought $\sigma$ is just a permutation on $A$ and we don't have $\sigma(x+y)=\sigma x + \sigma y$.

Edit--The question Lang's Algebra: Herbrand quotient has Lang's exercise written out fully with some slight corrections, but also the assumption $\sigma(x+y)=\sigma x + \sigma y$, so there is probably some typo or ambiguity in the definitions. I just wanted to make sure the homomorphism property didn't somehow follow from the $G$ being a finite cylclic group and/or $A$ being abelian.

facowal399
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  • What’s mean $G$ operates on $A$? – Federico Fallucca Feb 06 '20 at 21:48
  • "Operates on" here means that it acts via an endomorphism. See Wikipedia – Arturo Magidin Feb 06 '20 at 21:48
  • Where in the book is this statement? – rogerl Feb 06 '20 at 21:48
  • @FedericoFallucca it means a homomorphism from $G$ to the set of permutations on $A$ – facowal399 Feb 06 '20 at 21:49
  • @rogerl in my springer edition, it is question 45 of the first chapter. – facowal399 Feb 06 '20 at 21:50
  • @facowal399 so that map is only a permutation? Is not a morphism of algebra? – Federico Fallucca Feb 06 '20 at 21:51
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    Page 55 (in my edition), in section 11, defines the term "operation" as a homomorphism from $G$ into $\mathrm{Aut}(A)$. – rogerl Feb 06 '20 at 21:54
  • @rogerl I see that in my edition too. But there is an entire section called "Operations of a Group on a Set" where it is defined as a map into $Perm(A)$, page 25. I have not reached the later definition, perhaps that resolves my question though: The term "operation" is ambiguous? – facowal399 Feb 06 '20 at 21:57
  • @facowal399 in the linked question $\sigma(x+y) =\sigma(x) +\sigma(y) $ is assumed. – Berci Feb 06 '20 at 21:58
  • @Berci I see that at https://math.stackexchange.com/questions/195451/langs-algebra-herbrand-quotient and that question also has some additional slight terminology changes from Lang (eg endomorphism to homormophism). I am not sure where that question-asker got the changes from. Perhaps there is erratum, but I don't see it in the common online errata. – facowal399 Feb 06 '20 at 22:00
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    Indeed, the term "operation" is context-dependant. When we say "operates on an abelian group" it is implied that the operation is by automorphism of group. It it had said "operates on the underlying set of an abelian group $A$" then it would have meant that the operation is by simple permutations. – Captain Lama Feb 06 '20 at 22:04
  • @CaptainLama If you want to put it as an answer, I'm happy to mark it so – facowal399 Feb 06 '20 at 22:11

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