Download fifty problems

FIFTY PROBLEMS
Problem 1 If G is a group in which (ab)i = ai bi for three consecutive integers i for all a, b ∈ G, show that G is abelian.
Problem 2 Let p be a prime. Show that a group of size pn must have nontrivial center.
Problem 3 Let G be a group and let N be a normal subgroup of G. Show
that G is solvable if and only if G/N and N are solvable.
Problem 4 Let p and q be primes. Show that groups of size pn , pq, p2 q,
p2 q 2 are solvable. NOTE: p2 q 2 is not easy.
Problem 5 Show that any group of order less than 60 is solvable.
Problem 6 (Hard) Show that if G is a non-solvable group of order 60, then
G∼
= A5 .
Problem 7 Let G be a group and let Z denote the center of G. Show that
if G/Z is cyclic, then G = Z.
Problem 8 Show that a group of order p2 is abelian.
Problem 9 Let G be a finite group and let p be the smallest prime factor of
|G|. Show that a subgroup of index p is necessarily normal.
Problem 10 Prove Cauchy’s theorem as follows. Let G be a group and let
p be a prime dividing the order of G. Create the set
X := {(g1 , . . . , gp ) | g1 g2 · · · gp = 1, g1 , . . . , gp ∈ G} ⊆ Gp .
We let Z/pZ act on X as follows. Given an integer a with 0 ≤ a < p, we
define
(a + pZ) · (g1 , . . . , gp ) := (g1+a , g2+a , . . . , gp , g1 , . . . ga );
that is, the action is a cyclic permutation.
1. Show that the action described above is indeed an action on X.
2. Show that X has size |G|p−1 .
3. Show that there exist at least p elements of X which have orbits of size 1.
(Hint: consider the element (1, 1, . . . , 1) ∈ X.)
4. Conclude that there must exist an element g 6= 1 in G with g p = 1.
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Problem 11 Let G be an infinite group. Show that if G has a subgroup of
finite index, then G has a normal subgroup of finite index. In particular, G
is not simple.
Problem 12 Give an example of a non-abelian group all of whose subgroups
are normal.
Problem 13 Let G be a finite group and let H be a proper subgroup. Show
that
[
G 6=
gHg −1 .
g∈G
Give an example to show that equality can hold if G is infinite. (Hint: think
of invertible matrices over the complex numbers.)
Problem 14 Let A and B be two normal subgroups of a group G. Show that
if A ∩ B = {1}, then ab = ba for all a ∈ A and b ∈ B.
Problem 15 Show that if K ⊆ H ⊆ G are groups with H normal in G and
K characteristic in G, then K is normal in G. Show that if H is characteristic in G and K is normal in G that K need not be normal in G.
Problem 16 Let F be a finite field of size q = pm , p a prime. Let GLn (F )
denote the group of all invertible n × n matrices with entries in F . Show that
|GLn (F )| = (q n − 1)(q n − q) · · · (q n − q n−1 ).
Problem 17 How many automorphisms does
n
Y
Z/pZ
i=1
have? (Hint: use the preceding exercise.)
Problem 18 Give a p-Sylow subgroup of GLn (F ), where F is a finite field
of size q = pm .
Problem 19 Find the center of the following groups: Sn , Dn , GLn (F ).
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Problem 20 Show that
(C∗ , ×) ∼
= (R, +) × S 1 ,
where S 1 denotes the group of complex numbers of modulus 1 under multiplication.
Problem 21 (Medium) Let G be a finite group and suppose Φ : G → G is
an automorphism satisfying:
1. If Φ(x) = x then x = 1; and
2. Φ ◦ Φ(x) = x for all x ∈ G.
Show that G is abelian.
Problem 22 Prove that a finite group of size at least 3 has a non-trivial
automorphism.
Problem 23 Let G be a finite group of odd order equal to n. Let g1 , . . . , gn
be a listing of the elements of G. Show that g1 g2 · · · gn ∈ G0 ; that is, the
product of all elements of G will be in the commutator subgroup, regardless
of the order in which you multiply. Compare this result to Wilson’s theorem.
Problem 24 Let G be a group and let Z denote its center. Show that Z is
a characteristic subgroup of G.
Problem 25 Let G be a finite group and let F be a field. Show that G is
isomorphic to a subgroup of GLn (F ) for some n.
Problem 26 (Hard) Let S∞ denote the group consisting of all permuations
of {1, 2, 3, . . .} which fix all but finitely many natural numbers. Show that S∞
cannot be embedded in GLn (F ) for any field F and any natural number n.
Problem 27 Let p and q be primes with p < q. Show that if p does not
divide q − 1, then a group of order pq is cyclic. Show that if p | (q − 1) then
there exists a non-abelian group of order pq. Show, in addition, that any two
non-abelian groups of order pq are isomorphic.
Problem 28 Suppose G is a group of odd order and that g ∈ G is conjugate
to its inverse. Show that g = 1.
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Problem 29 (Useful formula) Let σ ∈ Sn be a permutation which, when
written as a product of disjoint cycles, has mi cycles of length i for
1 ≤ i ≤ n. Show that the conjugacy class containing σ has size
n!/(1m1 2m2 · · · nmn m1 ! m2 ! · · · mn !).
Problem 30 Let G be a finite group and suppose g ∈ G has exactly two
conjugates. Show that G has a non-trivial normal subgroup.
Problem 31 Find two elements in A5 which are conjagate in S5 but not in
A5 .
Problem 32 Let p be a prime number and let G be a group of size pn .
Suppose H is a proper subgroup of G. Show that NG (H) strictly contains H.
Show that G has a subgroup of size pm for 0 ≤ m ≤ n.
Problem 33 Calculate the size of a p-Sylow subgroup of Sn .
Problem 34 If G is a group of size 231, show that the 11-Sylow subgroup is
normal and that the 7-Sylow subgroup is in the center of G.
Problem 35 (Hard) Show that if G is a finite group with a p-Sylow subgroup
P lying in the center of G, then G ∼
= P × N for some normal subgroup N of
G.
Problem 36 Let p be a prime and let G be a finite group in which (ab)p =
ap bp for all a, b ∈ G. Show that a p-Sylow subgroup of G is normal in G and
that G ∼
= P × N for some normal subgroup N of G.
Problem 37 Show that a group of size 108 has a normal subgroup of size
3k , where k ≥ 2.
Problem 38 Find the possible number of 11-Sylow subgroups, 7-Sylow subgroups, and 5-Sylow subgroups of a group of size 52 · 7 · 11.
Problem 39 How many abelian groups of order 24 · 36 are there?
Problem 40 Show that (Z/pZ)∗ ∼
= Z/(p − 1)Z.
Problem 41 (Hard) Let G be a finite group. Suppose xn = 1 has at most n
solutions in G for each n ≥ 1. Show that G is a cyclic group. (Note: there
is no abelian hypothesis.)
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Problem 42 Show that the nonzero elements of a finite field form a cyclic
group.
Problem 43 Let G be a finite group and let P be a p-Sylow subgroup of G.
Show that NG (NG (P )) = NG (P ).
Problem 44 Show that G cannot be the set-theoretic union of 2 proper subgroups.
Problem 45 Show that G is a set-theoretic union of 3 proper subgroups if
and only if Z/2Z × Z/2Z is a homomorphic image of G.
Problem 46 Let G be a finite group and let {g1 , . . . , gn } be a complete set
of conjugacy class representatives for G. Show that if gi commutes with gj
for 1 ≤ i < j ≤ n then G is abelian.
Problem 47 Let p be a prime. Construct a nonabelian group of order p3 .
Problem 48 (Hard) Let G = SL2 (C), the 2 × 2 matrices of determinant one
with complex entries. Show that G0 = G.
Problem 49 Show that the group of automorphisms of C(t) which fix C
is isomorphic to SL2 (C)/Z, where Z = {I, −I} is the center of SL2 (C).
(For those who know some complex analysis.) Let P1 denote C ∪ {∞}, the
one point compactification of the complex numbers. What are the bijective
meromorphic functions from P1 to itself ? Can you explain the connection
between automorphisms of P1 and automorphisms of C(t)?
Problem 50 (This isn’t really related to 200C, but it is interesting nevertheless.) Show that π is irrational as follows.
1. Suppose π = a/b with a, b positive integers. Let f (x) = xn (bx − a)n /n!,
where n is some large integer which will not be specified at this point. Show
that f (k) (0), f (k) (π) are integers.
2. Let F (x) = f (x) − f (2) (x) + f (4) (x) + · · · + (−1)n f (2n) (x). Show that
F (x) + F 00 (x) = f (x) and that F (0), F (π) are integers.
3. Show that
(F (x) cos x)0 = −F (x) sin x + F 0 (x) cos x
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and
(F 0 (x) sin x)0 = F 0 (x) cos x + F 00 (x) sin x.
Conclude that
(F 0 (x) sin x − F (x) cos x)0 = f (x) sin x.
4. Using parts 2 and 3 show that
Z π
f (x) sin x dx ∈ Z.
0
5. Observe that f (x) and sin x is a continuous nonnegative on [0, π] that is
not identically zero. Hence
Z π
f (x) sin x dx.
0 <
0
6. Show that, given ε > 0, for large n, |f (x) sin x| < ε and hence for n
sufficiently large
Z π
f (x) sin x dx < 1.
0
Obtain a contradiction using part 5 and part 4. Conclude that π is irrational.
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