Mathematics 581   Problem Set 5   Due Monday, May 13

1. Let  G  be any group.  Let  f  be the function from  G  into  G  defined by f(a) = a'  for every  a  in  G.

(a)  Show that  f is  one-to-one and onto.

(b)  Let  H  be any subgroup of  G.  Show that  f  maps H, which is a subset of  G, onto  H.

(c)  Let  a  be any element of  G.  Show that  f  maps the left coset  aH, which is a subset of  G, onto the right coset  Ha¢.

2.  Let  f be an isomorphism of a group  G  onto a group   C.  Let  a  be an element of  G.  Show that the order of the element  f(a)  in  C  is equal to the order of the element  a. (Two cases:  order  a  is infinite, and order  a  is a finite positive integer  n.)

3.  This is a generalization of problem  2.  Let  be a homomorphism of a group  G  into a group   C.  Let  a  be an element of  G  and let the order of  a  equal n,  finite.  Show that the order of  f(a)  is finite and a factor of  n.

4. (a)  Show that if   s   and  t  are disjoint cycles then  st  =  ts.

(b)  Show that the order of the permutation  (1 2 3)(4 5) is 6.
(c)  Show that the order of  (1 2 3 4)(5 6 7)  is  12.
(d)  What is the order of  (1 2)(3 4 5)(6 7 8 9)? (It is not 24.)
(e)  What is the order of  (1 2 3 4 5 6)(11 12 13 14 15 16 17 18 19 20)?
(f)  Prove that if a permutation  in Sn  can be written as a product of disjoint cycles s and the cycles have lengths   m1, m2, ..., mk, then the order of  is the least common multiple of  m1, m2, ..., mk.

5.  Put  a Cartesian coordinate system onto the plane.  Let F  be the geometric figure in the plane which consists only of the points along the x-axis with  integer coefficients.  That is,  F =  { ..., (-2,0), (-1, 0), (0, 0), (1, 0), (2, 0), ...}. These points are spaced 1 unit apart.  Remember that a symmetry of  is nothing more than a one-to-one map of  onto  F  which preserves distances.   Find all of the symmetries of  F.  (Hint:  think of  F  as like a regular polygon with infinitely many sides extending to the left and right of any starting point.  Think of what might be analogues for  F  of rotations and reflections  on  an  n-gon.)