MATH 417 Study Guide - Midterm Guide: Inclusion Map, Abelian Group, Trivial Group
16 Oct 2018
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For each of the following statements indicate whether it is true or false. Answers. (1) true. (2) false. (the statement would become true if in addition we assume that the (3) true. Apply the first isomorphism theorem to the map : c r>0, for g = z2 z2 and for n = 4 we have n(cid:12)(cid:12)|g| but g has no function is a homomorphism. ) H = g gives g/h = {eh}, the trivial group, which is nitely generated. [10 points] for any group g (even a non- nitely generated group), taking. For each of the following pairs of groups indicate whether or not the groups in this pair are isomorphic. Solution. (3) we have z4 z3 (1) we have sl(2, z) (cid:54) = z z because z z is abelian but sl(2, z) is non- abelian. (2) we have (r, +) (cid:54) = (r , ). Indeed, the group (r , ) has an element of order.
