Math 366–001 HW 3, Spring 2015
This assignment is due Friday, February 20 in class. Feel free to work together, but be
sure to write up your own solutions.
As for writing it up, please write legibly on your own paper, including as much justification
as seems necessary to get the point across.
1. Find the inverse of the element
2 6
3 5
in GL(2, Z11 ). (Notice that we are working
over Z11 , not R!)
2. Complete the following Cayley table for some group G:
e a b c d
e e
a
b
e
c d e
b
c
d
a b
d
3. Find the order of each of the following five groups:
(a) (Zn , +)
(b) (Z×
n , ∗)
(c) (U (10), ∗)
(d) (U (20), ∗)
(e) D3
4. What are the orders of all twelve elements of the additive group Z12 ? (No need to
answer the following, but do you notice anything about their orders compared to the
order of the group, other than that they are less than or equal to 12?)
5. Let x ∈ G (a multiplicative group). If x2 6= e and x6 = e, then prove that x4 6= e and
x5 6= e.
6. Suppose that H is a proper subgroup (where “proper” means “smaller than the whole
group”) of the additive group Z and 18, 30, 40 ∈ H. What is H?
7. If a and b are distinct elements of a multiplicative group, prove that you cannot have
both a2 = b2 and a3 = b3 .
More on the back!!!
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8. Find the orders of both A =
0 −1
1 0
and B =
0
1
−1 −1
in SL(2, R) (an infinite
group!).
9. Consider the infinite group D∞ of symmetries of the circle. Describe the set of rotations
of finite order.
10. For D∞ , describe the set of rotations of infinite order.
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