# Math 441 Assignment 8 Solutions

1. [2 points] List the elements of the group U _{20}.

U_{20} is the set of units in Z_{20}, which are the elements [a] ∈Z_{20}
which are relatively

prime to 20. Then

2. (a) [2 points] Show that the composition of two injective functions is
injective.

Suppose g : A -> B and f : B -> C are injective functions. Then suppose

for some
Then
and

since f is injective, that means that
Since g is also injective,

this means that a_{1} = a_{2}.

Then the composition is injective.

(b) [2 points] Show that the composition of two surjective
functions is surjective.

Suppose g : A -> B and f : B -> C are surjective functions, and let

c ∈C. Since f is surjective, there is some b ∈B such that f(b) = c.

Since g is surjective, there is some a ∈A such that f(a) = b. Then

Then the composition is surjective.

(c) [4 points] Let T be a nonempty set and let A(T) be the
set of all permutations of

T. Show that A(T) is a group under the operation of composition of functions.

(You may assume that a bijective function has an inverse that is bijective.)

A permutation in A(T) is a bijection (injective and surjective function)

from T to T.

•A(T) is closed under composition: Let f : T -> T and g :
T -> T

be two permutations in A(T). Then both f and g are bijective and

by parts (a) and (b), their composition : T
-> T is a bijective

function. So A(T) is closed under composition.

•Composition of functions is associative: Let f, g, h ∈A(T).
Then, for

any t ∈T,

•The identity element in A(T) is the identity function
id(t) = t for all

•If f is in A(T), then f is a bijective function from T to T and has

an inverse f^{-1} from T to T which is also a bijective function. So

for all

t∈T. So for any f∈A(T), there is an element f^{-1}∈A(T) such that

3. [10 points] Let 1, i, j, k be the following matrices with complex entries :

Show that **Q = {1, i,-1,-i, j, k,-j,-k}** is a group
under matrix multiplication by

writing out its multiplication table . Q is called the quaternion group.

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