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A companion to S.Lang's Algebra 4ed. by Bergman G.M.

By Bergman G.M.

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This group, which as noted above is called the group ‘‘presented by generators S and relations T ’’, is often written S T or S R by group-theorists. ) Note that to show that a given group G has presentation S T , one must verify not only that the G. M. 42 generators S satisfy the relations T in G, but that the relations T imply all the relations satisfied by S in G. We do not have time in this course to examine how this is done in various situations (it can depend on the sense in which G is ‘‘given’’), nor the reverse problem of obtaining concrete descriptions of groups presented by generators and relations.

Whenever the universal property of free abelian groups is being discussed, the concept in question is the one involving a specified basis. In the immediately following passage in Lang, on products and coproducts, you will encounter a similar situation. The product P of two objects A and B (to take the case you have some familiarity with) will be defined to be an object given with morphisms f, g to A and B respectively, called ‘‘projections’’, such that P is universal among all objects C given with two morphisms ϕ : C → A and ψ : C → B.

Some examples: If is the category of vector spaces B over a field k, and B is the 1-dimensional space k, then h is the ‘‘dual vector space’’ construction. The behavior of this construction with respect to linear maps probably constitutes the first example of contravariance most students see, though, of course, they do not see it under that name. Here the hom-sets can be given structures not only of sets, but of vector spaces, giving us a functor from vector spaces to vector spaces. (The circumstances under which the sets arising as values of a representable functor, covariant or contravariant, acquire algebraic structure, are something that we won’t be able to cover here.

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