MATH 433
April 10, 2015
Quiz 9: Solutions
Problem 1. The multiplicative group GL(2, Z2 ) has 6 elements
[0] [1]
[1] [1]
[1] [0]
[1] [1]
[0] [1]
[1] [0]
,
,
,
,
,
[1] [1]
[1] [0]
[1] [1]
[0] [1]
[1] [0]
[0] [1]
(where [0] and [1] are congruence classes modulo 2). Find the order for every element of this group.
[1] [0]
[1] [1]
[0] [1]
[1] [0]
have order 2,
, and
,
has order 1,
Solution:
[1] [1]
[0] [1]
[1] [0]
[0] [1]
[0] [1]
[1] [1]
have order 3.
and
[1] [1]
[1] [0]
[1] [0]
The identity element of the group GL(2, Z2 ) is the matrix I =
. It is the only element of
[0] [1]
order 1. The orders of the other elements are found by direct multiplication:
[1] [1]
[1] [0]
2
[0] [1]
[1] [1]
2
[0] [1]
[1] [0]
2
=
[0] [1]
[1] [1]
=
[1] [1]
[1] [0]
2
=
6= I,
[1] [1]
[1] [0]
3
6= I,
[0] [1]
[1] [1]
3
=
[1] [1]
[0] [1]
[1] [0]
[1] [1]
2
= I;
=
[1] [1]
[1] [0]
[0] [1]
[1] [1]
= I;
=
[0] [1]
[1] [1]
[1] [1]
[1] [0]
= I.
Problem 2. List all cyclic subgroups of the group Z15 .
Solution: {[0]}, {[0], [5], [10]}, {[0], [3], [6], [9], [12]}, and Z15 .
Every congruence class [a]15 generates a cyclic subgroup of Z15 , h[a]15 i = {[na]15 | n ∈ Z}. We obtain
that
h[0]i = {[0]},
h[5]i = h[10]i = 5Z15 = {[0], [5], [10]},
h[3]i = h[6]i = h[9]i = h[12]i = 3Z15 = {[0], [3], [6], [9], [12]},
h[1]i = h[2]i = h[4]i = h[7]i = h[8]i = h[11]i = h[13]i = h[14]i = Z15 .