MAT401H1 Lecture Notes - Lecture 1: Coset, Abelian Group, Cyclic Group
Document Summary
Assignment 1 solutions: [8] determine if each statement is true or false. Solution: the statement (a) is false and the rest are all true: [5] suppose : g h is a group homomorphism. Let g g be an element of nite order. Thus (g) is of nite order and | (g)| (cid:12) (cid:12) n. (make sure you know why this follows. ) = 1h: let g be a group. 2 (c) [5] recall that the dihedral group d2n is the group of symmetries of a regular polygon with n sides. It has 2n elements: n rotations and n re ections. Is d2n/r abelian? (d) [3] determine if the following statement is true or false. If it is false, give a counter- example, and if it is true, prove the statement. If k, h are normal subgroups of a group g such that k h, and g/h and h/k are both abelian, then g/k is also abelian.
