C Roettger
Spring 2013
Math 301 – Homework 7
Problem 9.16 Recall that Z(D6 ) = {R0 , R180 }. What is the order of the
element R60 Z(D6 ) in the factor group D6 /Z(D6 )?
Problem 9.26 Let G be the group of quaternions given by the table in Exercise 4 of the Supplementary Exercises for Chapter 1-4 on page 91. For your
convenience, the table is below. I have changed the notation a bit – I hope you
see how it becomes simpler. In particular, you only really need to know the top
left quarter of the table. We use multiplication for the binary operation, and
list the elements of G as
G = {1, i, j, k, −1, −i, −j, −k}
Here, we have an element named −1. It got its name because it behaves exactly like −1 under multiplication, eg (−1)i = i(−1) and (−1)2 = 1. We use
−i as shorthand for (−1)i and so on. The rules of the multiplication can be
summarized as i2 = j 2 = k 2 = −1, ij = −ji, and ij = k, jk = i, ki = j.
1
·
1
1
i
i
j
j
k
k
−1 −1
−i −i
−j −j
−k −k
i
i
−1
−k
j
−i
1
k
−j
j
k
j
k
k −j
−1
i
−i −1
−j −k
−k
j
1 −i
i
1
−1 −i
−1 −i
−i
1
−j
k
−k −j
1
i
i −1
j −k
k
j
−j −k
−j −k
−k
j
1 −i
i
1
j
k
k −j
−1
i
−i −1
Let H be the subgroup {1, −1}. Is G/H isomorphic to Z4 or to Z2 ⊕ Z2 ?
Problem 9.32 Prove that D4 cannot be expressed as an internal direct product
of two proper subgroups.
Problem 9.34 In Z, let H = h5i and K = h7i. Prove Z = HK. Is it true that
Z = H × K? If yes, prove it. If not, why not?