Mathematics 466
Final (50 points)
Due 12/18/08, 11.10am
Points (leave blank)
1
2
3
4
5
∑
Name:
(clearly, please)
This exam is my own work. Sources (apart from the textbooks and my lecture notes) are indicated.
Signature
Notes
● Put your name on this cover sheet and sign it.
● The exam must be given to me, or put in my mailbox (via the front office) by the indicated
time.
● You are permitted to use your notes and any publication (book, journal, web page). You are
not permitted to consult persons.
Results which are quoted from a publication (apart from the course textbooks and your lecture
notes) should be indicated.
● Unless specified otherwise, you may use a computer for arithmetic.
Computer commands, which render the problem trivial (for example using factorization to
show that a polynomial is irreducible over Z), however are not considered a sufficient solution.
● If you need extra sheets, staple them to this exam. (You do not need to submit scrap paper.)
1) For a group G, we denote by C the set of its conjugacy classes.
a) The center of a group G is the set
Z(g) = {g ∈ G ∣ ag = ga∀a ∈ G}
of elements of G commuting with all other elements of G. Show that Z(G) ⊲ G and that
Z(G) =
⋃
C
C∈C,∣C∣=1
is the union of all conjugacy classes of G which have order 1.
b) Now let p be a prime and G a group with ∣G∣ = p a . Using that ∣G∣ = ∑C∈C ∣C∣, show that ∣Z(G)∣ > 1.
(Hint: ∣C∣ is a divisor of ∣G∣, the class of the identity element has size 1.)
c) Let ∣G∣ = p2 with ∣Z(G)∣ = p. Show that there must be an element x ∈ G with x ∈/ Z(G), that
G = ⟨Z(G), x⟩ and that G is abelian.
d) Show that every group of order p2 is abelian.
2) Suppose that G acts on Ω and x ∈ Ω. Let S = StabG (x).
a) Show that for g ∈ G we have that g −1 S g is the stabilizer of µ(x, g) in G.
b) Show that if x, y are in the same orbit of G and N ⊲ G such that N ≤ StabG (x), then N ≤ StabG (y).
c) Suppose that G acts transitively (i.e. there is one orbit) on Ω and θ∶ G → S Ω is the corresponding
action homomorphism. Show that if N ⊲ G stabilizes one element of Ω, then N ≤ ker θ.
3) Let G = S5 and Ω the set of all subsets of {1, . . . , 5} of cardinality 2. (So ∣Ω∣ = (25) = 10.) Consider
the action of G on Ω via µ({a, b}, g) ∶= {µ(a, g), µ(b, g)}. Let θ∶ S5 → S10 be the homomorphism
defined by this action.
(In GAP, you can create this homomorphism as
theta:=ActionHomomorphism(G,Combinations([1..5],2),OnSets);
a) Let H = Gθ. Show that H ≅ G and determine representatives and the cardinality of the conjugacy
classes of H.
b) Consider the set of complete graphs on the vertices 1 to 5 with edges colored black or white. (This
is a way of representing all graphs on 5 vertices.) Permuting the vertices wil implicitly permute the
edges. If we label each edge with the set of its two vertices, a vertex permutation g induces the edge
permutation gθ. Thus H is the group of symmetries of these colored graphs.
Determine the number of orbits of H on these colored graphs. (This is the same as the number of
different graphs on 5 vertices.)
4) Let G ≤ S n be a permutation group and F be a field such that 1 =/ −1 in F. We consider the
n-dimensional row space V = F n .
a) Show that G acts on V by permuting the indices, i.e.
µ ((a1 , . . . , a n ), g) ∶= (a1(g−1 ) , a2(g−1 ) , . . . a n(g−1 ) )
(The inverse occurs as the permutation moves the entry in position 1 to position 1 g , thus what will
be in position 1 is what was in position 1( g −1 ).)
b) Show that for every g ∈ G the map Tg ∶ V → V , v ↦ µ(v, g) is a linear transformation. We assign
to it a matrix M g , representing this transformation with respect to the standard basis.
c) Show that the map θ∶ G → GLn (F), g ↦ M g is a group homomorphism.
d) Show that the map σ∶ G → F, g ↦ det(M g ) is a group homomorphism. Show that the image of σ
is {1, −1}. We call g σ the sign of the permutation.
e) Show that g σ = −1k , where k is the number of cycles of the permutation g of even length.
f) Show that the permutations of sign 1 form a normal subgroup of S n of index 2. (This subgroup is
called the alternating group.)
5) Let G be a finite group, N ⊲ G and P ≤ N a p-Sylow subgroup of N.
a) Show that for g ∈ G we have that P g = g −1 Pg is a p-Sylow subgroup of N.
b) Conclude from this, that G = ⟨N , NG (P)⟩. (Hint: Show that there is n ∈ N, such that P g = P n .)