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I am reading Paolo Aluffi's book, Algebra: Chapter 0.
I am studying Chapter II: Groups, first encounter. In Section 3 of this chapter, Aluffi deals with the category Grp in which the objects are groups and the morphisms are group homomorphisms.
Section 3,3, which is a pause for reflection, reads as follows:View attachment 2668
I am somewhat confused by the specific function described in that Aluffi writes:
$$ i_G \: \ G \to G , \ \ i(g) := g^{-1} $$.I have two rather simple questions:
1. Why do we have $$ i_G $$ in one place and $$ i $$ in the other - that is, shouldn't the above read $$ i_G \: \ G \to G , \ \ i_G(g) := g^{-1} $$?
2. Aluffi mentions both the identity element and inverses, but the function he considers seems to only deal with inverses? What is going on?
I realize that these are pretty simple issues, but would appreciate someone clarifying the situation for me.
Peter
***EDIT*** I have been reflecting on the above and now feel, regarding question 2 above, that I misunderstood what Aluffi was saying - the function specified only referred to inverses.
I am studying Chapter II: Groups, first encounter. In Section 3 of this chapter, Aluffi deals with the category Grp in which the objects are groups and the morphisms are group homomorphisms.
Section 3,3, which is a pause for reflection, reads as follows:View attachment 2668
I am somewhat confused by the specific function described in that Aluffi writes:
$$ i_G \: \ G \to G , \ \ i(g) := g^{-1} $$.I have two rather simple questions:
1. Why do we have $$ i_G $$ in one place and $$ i $$ in the other - that is, shouldn't the above read $$ i_G \: \ G \to G , \ \ i_G(g) := g^{-1} $$?
2. Aluffi mentions both the identity element and inverses, but the function he considers seems to only deal with inverses? What is going on?
I realize that these are pretty simple issues, but would appreciate someone clarifying the situation for me.
Peter
***EDIT*** I have been reflecting on the above and now feel, regarding question 2 above, that I misunderstood what Aluffi was saying - the function specified only referred to inverses.
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