2. Find all subgroups of Z_{11}^{×}, and give the lattice diagram which shows the inclusions between them. Solution
3.
Let G be the subgroup of GL_{3} (R)
consisting of all matrices of the form
such that a and b are in R.
Show that G is a subgroup of GL_{3} (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 Z_{20}^{×}.
Is the factor group
Z_{20}^{×} / < 9 > cyclic?
Use the notation Z_{20}^{×}
= { ± 1, ± 3, ± 7, ± 9 }.
Solution
6.
Let G be the subgroup of GL_{2} (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
D_{4} = { e,a,a^{2},a^{3},b,ab,a^{2}b,a^{3}b },
where a^{4} = e, b^{2} = e, and b a = a^{3} b. Let N be the subgroup < a^{2} > = { e, a^{2} }.
8.
Let G = D_{8},
using notation similar to Problem 7 above,
and let N = { e,a^{2},a^{4},a^{6} }.
(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