Math 211 Final Exam
(Group Theory)
Ali Nesin
1. Let n > 1 be an integer and let m be a divisor of n.
a) Show that /n has m elements whose order divides m. (5 pts.)
Answer: Let x ∈ be such that mx = 0 in /n. Then n mx and hence n/m x.
Therefore x ≡ 0, n/m, 2n/m, ..., kn/m, ..., (m−1)n/m. Thus there are exactly m
elements in /n whose order divides m.
b) How many elements does /n have whose order is exactly m? (5 pts.)
Answer: By above, {x ∈ /n : mx = 0} = 〈 n/m 〉 ≈ /m. Thus number of
elements of /n whose order is exactly m is ϕ(m).
c) Show that a subgroup of a cyclic group is cyclic. (5 pts.)
Answer: Let G be a cyclic group. If G ≈ , this must be clear at this stage of your
studies. Assume G ≈ /n for some n > 1. Let H ≤ /n have order m. Then mn
and H ≤ {h ∈ /n : mh = 0}. By part a, H = {h ∈ /n : mh = 0} and H is cyclic.
2. Let p be a prime. Suppose that G has a normal and nontrivial p-subgroup. Show that G
has a normal and nontrivial abelian subgroup. (5 pts.)
Answer: Let H G be a p-subgroup. Since Z(H) ≠ 1 and Z(H) is characteristic in H, Z(H)
G.
3. Let p ≤ q be two primes and G a group of order pq. Show that if q 1 mod p then G is
abelian. (5 pts.)
4. Let p < q be two primes and G a group of order pq
2
. Show that if q ±1 mod p then G
is abelian. (10 pts.)
5. Let p < q be two primes with q ≡ 1 mod p.
a) Show that there are at most p nonisomorphic groups of order pq. (5 pts.)
b) Show that the upper bound p may be attained. (5 pts.)
6. Let p and q be two primes such that q 1 mod p. Let n be a natural number. Let G be a
group of order p
n
q.
a) Show that G has a nontrivial normal abelian subgroup. (5 pts.)
b) Show that there is a sequence 1 = G
0
≤ G
1
≤ ... ≤ G
n
≤ G
n+1
of normal subgroups such
that G
i+1
/G
i
is of prime order for each i = 0, ..., n. (5 pts.)
7. Give the correct mathematical definition of the following “definition”: A finite group G
is called solvable if either G = 1 or there is a nontrivial normal abelian subgroup A such
that G/A is solvable. (5 pts.)
8. a) Show that if H
1
and H
2
are two subgroups of finite index of the group G, then H
1
∩
H
2
is a subgroup of finite index of G. (5 pts.)
b) Show that if H is a subgroup of finite index of the group G, then H has finitely many
conjugates. (5 pts.)
c) Show that if H is a subgroup of finite index of the group G, then there is a normal
subgroup N of finite index in G such that N ⊆ H. (5 pts.)
9. Let H ≤ G be a subgroup of finite index, say n. Let X = G/H = {xH : x ∈ G} (the left
coset space). For g ∈ G and xH ∈ X, define ϕ
g
(xH) = gxH.
a) Show that ϕ
g
is a bijection of X, so that ϕ
g
∈ Sym(X). (2 pts.)
