MATH 518
Homework #4
Dr. Carroll
Peter Larson
1. For G as in exercise #4 in Chapter 4:
a. Find the order of each of the eight elements.
Element
order
1
1
y
4
2
2
y3
x
xy
4
y
4
4
xy
2
4
xy
3
4
b. How many elements of order 2 are there? 1
c. Explain why every proper subgroup must be cyclic.
Every proper subgroup of G will contain the powers of x and y and hence be cyclic. Every
element of any proper subgroup of G can be generated from a single element.
2. Let G be an abelian group where a, b  G such that a  e and b  e . Describe the
2
3
elements in <a,b>.  {e, a, b, b , ab, ab }
2
2
3. Let G and H be groups and A  G, B  H . Show AxB  GxH .
GxH is the set of all ordered pairs (g,h) with g  G and h  H , i.e.GxH= {( g , h) | g  G, h  H } .
Let a  A and b  B . Then (a,b) is in GxH and also in AxB. Since A and B are subsets of G a H
respectively, and since every ordered pair consisting of elements of A and elements of B will only
be in AxB and that each of those ordered pairs is also in GxH, we can conclude that
AxB  GxH .
4. Let G be any group and let H= {g  G | xg  gxx  G} . Prove that H is a subgroup of G.
a. The inverse of G is in H. xe=ex.
b. If a, b  H then xa=ax and xb=bx. So a(xb)=(ax)b=a(bx) , substituting xa for ax,
xab=abx so that ab is also in H.
c. For any g  H , xg  gx  gxg 1  g 1 gx  x  g 1 gxg 1  xg 1  g 1 x so that the
inverse of any element in H is also in H.
Hence, H is a subgroup.