Universidad de los Andes
Álgebra Abstracta · Examen Final
Miércoles, 13 Mayo 2015, 5 questions and 30 marks
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Last Name
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• Write your names in the blank spaces above
• Show all your work and fully explain the process to receive full credit
Q
mark
1
2
3
4
5
total
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1+1+2+2 marks
1. Let H and K be groups and let ϕ : K → Aut(H) be a homomorphism.
(a) Write the definition of the operation in the semidirect product H oϕ K. What is the order of the
semidirect product?
(h1 , k1 ) · (h2 , k2 ) =
(b) What can you say about the group structure when K is normal in H oϕ K?
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(c) Consider V4 = {1, a, b, c} ' Z2 × Z2 and the isomorphism S3 → Aut(V4 ). In the holomorph
Hol(V4 ) ' V4 o S3 , compute the inverse of (b, σ) and its order, where
x
1
a
b
c
σ(x)
1
c .
a
b
(b, σ)−1
=
|(b, σ)| =
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(d) How many groups of order 6 are there? Write each one as a semidirect product of smaller groups.
Write down the entire construction explicitly.
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4+1+1 marks
2. Consider the quasidihedral group of order 16 with the presentation
QD16 = hσ, τ : σ 8 = τ 2 = 1, στ = τ σ 3 i.
A partial lattice appears on the next page. The three maximal subgroups have order 8. Fill in the blanks.
Represent each subgroup by exhibiting at most two generators. Is QD16 solvable? Is it nilpotent?
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cada 1
3. True/False? Give brief explanations.
( T / F ) If G is abelian, then Hom(G, G) and Aut(G) are both abelian.
( T / F ) If |G| = 12 and G doesn’t have a normal Sylow 3-subgroup, then G is simple.
( T / F ) Every normal subgroup is the kernel of a homomorphism.
( T / F ) The terms {Zi (G)}i of the upper central series are all characteristic subgroups.
( T / F ) Let p and q, p < q, be primes. There are more abelian groups of order q n than abelian groups of
order pn .
( T / F ) The only finite abelian simple groups are the cylic groups Z/pZ where p is prime.
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2+1+5 marks
4. Answer the following questions about the Sylow theory.
(a) State the Sylow theorem.
(b) What can you say about a group that has only one Sylow subgroup?
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(c) Let G be a group of order 105 = 3 · 5 · 7. Show that if the Sylow 3-subgroup is normal, then G is
abelian.
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2+2 marks
5. Let R be a commutative ring. An element x ∈ R is called nilpotent if xn = 0 for some positive integer
n.
(a) Show that the set of all nilpotent elements, denoted N(R), is an ideal of R.
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(b) Compute N(Z/72Z).