Download Practice Problems for Math 370 Midterm

Practice Problems for Math 370 Midterm
1. How many homomorphisms are there from Z/2Z to the quaternion group Q?
2. True or false: If α : Z/3Z → Z/45Z and β : Z/45Z → Z/3Z are homomorphisms of
groups, then the composition β◦ : Z/3Z → Z/3Z is the trivial homomorphism. (Either
give a proof or a counter-example.)
3. True or false: Let H be a subgroup of S3 with 2 elements. Then there is a left H-coset
which is also a right H-coset. (Either give a proof or a counter-example.)
4. Give an example of a finite non-commutative ring.
5. Give an example of an infinite non-commutative group.
6. Give an example of a two linear transformations α, β : V → V of a vector space V such
that α ◦ β = 0 and β ◦ α 6= 0.
7. True or false: If G is a group with 12 elements, and H is a group with 15 elements,
and α : H → G is a non-trivial homomorphism of groups. Then the kernel of α is a
cyclic group with 5 elements.
8 Let V be a 10-dimensional vector space over the finite field F3 and let W be a 5dimensional vector subspace of W . How many element does V /W have? (Give a
complete proof.)
9. Let H be the subgroup of GL3 (R) consisting of all diagonal invertible matrices


a 0 0
 0 b 0 
0 0 c
with a, b, c ∈ R× . Is H a normal subgroup of GL3 (R)?
10. (extra credit) Recall that the group ring C[Z/3Z] is the ring consisting of all formal
linear combinations of the form
X
ax [x]
with ax ∈ C ∀x ∈ Z/3Z
x∈Z/3Z
and the multiplication is defined by

 

X
X

ax [x] · 
by [y] =
x∈Z/3Z
y∈Z/3Z
1
X
x,y∈Z/3Z
ax by [x + y] .
Moreover C[Z/3Z] has a natural structure as a C-vector space; the elements
{ [x] : x ∈ Z/3Z }
form a basis of C[Z/3Z]. For every element x ∈ Z/3Z, the map
Tx : C[Z/3Z] −→ C[Z/3Z]
u 7→ [x] · u
is a linear endomorphism of the C-vector space C[Z/3Z]. Write down the matrix
representations of the three Tx ’s (for x = 3Z, 1 + 3Z and 2 + 3Z). (A complete proof
is required for full credit.)
11. (extra credit) Find all group homomorphisms from the symmetric group S3 to the
cyclic group Z/6Z. How many homomorphisms are there? (A complete proof is
required for full credit.)
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