Practice Problems for Math 370 Midterm
1. How many homomorphisms are there from Z/2Zto the quaternion group Q?
2. True or false: If α:Z/3Z→Z/45Zand β:Z/45Z→Z/3Zare homomorphisms of
groups, then the composition β◦:Z/3Z→Z/3Zis the trivial homomorphism. (Either
give a proof or a counter-example.)
3. True or false: Let Hbe a subgroup of S3with 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→Vof a vector space Vsuch
that α◦β= 0 and β◦α6= 0.
7. True or false: If Gis a group with 12 elements, and His a group with 15 elements,
and α:H→Gis a non-trivial homomorphism of groups. Then the kernel of αis a
cyclic group with 5 elements.
8 Let Vbe a 10-dimensional vector space over the finite field F3and let Wbe a 5-
dimensional vector subspace of W. How many element does V /W have? (Give a
complete proof.)
9. Let Hbe the subgroup of GL3(R) consisting of all diagonal invertible matrices
a0 0
0b0
0 0 c
with a, b, c ∈R×. Is Ha 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
x∈Z/3Z
ax[x] with ax∈C∀x∈Z/3Z
and the multiplication is defined by
X
x∈Z/3Z
ax[x]
·
X
y∈Z/3Z
by[y]
=X
x,y∈Z/3Z
axby[x+y].
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