such that a and b are in R.
Show that G is a subgroup of GL3 (R). Solution
2. Find all subgroups of Z11×, and give the lattice diagram which shows the inclusions between them. Solution
3.
Let G be the subgroup of GL3 (R)
consisting of all matrices of the form
such that a and b are in R.
Show that G is a subgroup of GL3 (R).
Solution
4. Show that the group G in Problem 3 is isomorphic to the direct product R × R. Solution
5.
List the cosets of the cyclic subgroup
< 9 > in Z20×.
Is the factor group
Z20× / < 9 > cyclic?
Use the notation Z20×
= { ± 1, ± 3, ± 7, ± 9 }.
Solution
6.
Let G be the subgroup of GL2 (R)
consisting of all matrices of the form
,
and let N be the subset of all matrices of the form
.
(a) Show that N is a subgroup of G, and that N is normal in G.
(b) Show that the factor group G / N is isomorphic to
the multiplicative group R×.
Solution
7. Assume that the dihedral group of order 8 is written as
D4 = { e,a,a2,a3,b,ab,a2b,a3b },
where a4 = e, b2 = e, and b a = a3 b. Let N be the subgroup < a2 > = { e, a2 }.
8.
Let G = D8,
using notation similar to Problem 7 above,
and let N = { e,a2,a4,a6 }.
(a) List all left cosets and all right cosets of N,
and verify that N is a normal subgroup of G.
(b) Show that the factor group G / N has order 4, but is not cyclic.
Solution
Solutions to the problems | Forward to §4.1 | Back to §3.8 | Up | Table of Contents