Download GROUP THEORY 1. Groups A set G is called a group if there is a

GROUP THEORY
1. Groups
A set G is called a group if there is a multiplication (a group operation) G×G → G,
(g, h) 7→ gh such that
(1)
(∀f, g, h ∈ G) f (gh) = (f g)h,
(∃ e ∈ G)(∀g ∈ G) eg = ge = g,
(2)
(∀g ∈ G)(∃ g −1 ∈ G) gg −1 = g −1 g = e.
(3)
The condition (1) is called the associativity. The element e in (2) is called the
identity element, or the neutral element. It is easy to prove that such an element is
unique. The identity element is frequently denoted simply by 1. The element g −1
in (3) is called the inverse element. For any g ∈ G, the element g −1 is unique. Note
that the additive notation is used frequently instead of the multiplicative notation
introduced above. In the additive notation, the group operation is denoted by +,
the neutral element is denoted by 0 and is called zero, the inverse element of g is
denoted by −g and is said to be the opposite element. The trivial group consisting
of 1 (or 0) only is denoted by 0. A group G is called abelian (or commutative) if
xy = yx for all x, y ∈ G. Usually the additive notation is used for abelian groups.
In a non-abelian group, we say that two elements x and y commute if xy = yx.
Let G and H be groups. A map f : G → H is called a homomorphism if
f (gh) = f (g)f (h) for all g, h ∈ G. One can easily verify that f (1) = 1 and
f (g −1 ) = f (g)−1 . A homomorphism f is an isomorphism (resp., epimorphism,
monomorphism) if f is bijective (resp., surjective, injective). An isomorphism of
G with itself is called an automorphism of G. Take an element g0 ∈ G. The map
g 7→ g0−1 gg0 is an automorphism of G, which is called an inner automorphism, or
a conjugation. Elements g and g0−1 gg0 are called conjugate elements. Subgroups H
and g0−1 Hg0 are called conjugate subgroups.
A subset H ⊆ G is called a subgroup, if H is a group with respect to the multiplication inherited from G. A subset H ⊆ G is a subgroup iff gh−1 ∈ H for all
g, h ∈ H. For any homomorphism f : G → F , define the kernel Ker(f ) = f −1 (1)
and the image Im(f ) = f (G). The kernel Ker(f ) is a subgroup of G and the image
Im(f ) is a subgroup of F .
Let H be a subgroup of G. The set aH = {ah| h ∈ H} is called a (left) coset of
G with respect to H. The left cosets are the equivalence classes by the equivalence
relation x−1 y ∈ H. Therefore, cosets are disjoint. The set of all left cosets is denoted
by G/H. A right coset is a set of the form Ha, a ∈ G. The set of all right cosets
is denoted by H\G. The number of all left cosets of G with respect to H is called
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the index of H and is denoted by (G : H). The number of all right cosets coincides
with (G : H).
We say that a group G is generated by a set X ⊂ G if G is the only subgroup
containing X. A group is called finitely generated if it is generated by some finite
subset.
Problems and theorems.
1.1. The sets Z, Q, R, C are groups with respect to addition, and the sets Q∗ = Q − 0, R∗ =
R − 0, C∗ = C − 0 are groups with respect to multiplication.
1.2. Let X be an arbitrary set. The set of all bijections X → X is a group with respect to
composition. This group is called the transformation group of X.
1.3. The set Aut(G) of all automorphisms of G is a group with respect to composition.
1.4. A homomorphism f : G → F is injective iff Ker(f ) = 0, the homomorphism f is surjective iff
Im(f ) = F .
1.5. Prove that if H is a subgroup of G and F is a subgroup of H, then F is a subgroup of G and
(G : F ) = (G : H)(H : F ).
In particular, (G : H)|H| = |G|. Here |H| denotes the order of H, i.e. the number of elements in
H.
1.6. A group G is said to be cyclic if there is a an element g ∈ G s.t. G consists of powers of g.
Prove that any subgroup of Z is cyclic. Moreover, any subgroup of a cyclic group is cyclic. If G is
cyclic and f : G → H is an epimorphism, then H is also cyclic.
1.7. The Lagrange theorem. Let G be a group and g ∈ G. Denote by n the smallest positive
integer s.t. g n = 1 (if there is no such number, set n = ∞). The number n is called the order of
g. The order of an element g ∈ G divides the order |G| of the group G.
1.8. For arbitrary subsets H, F ⊆ G define F H = {f h ∈ G| f ∈ F, h ∈ H}. Give an example of
two subgroups H and F such that F H is not a subgroup. Prove that F H is a subgroup if and
only if F H = HF .
1.9. Suppose that xn = 1 for any x ∈ G. Then |G| divides a power of n.
1.10. Let p be a prime number. Prove that any group of order p is cyclic.
1.11. If two cyclic groups have the same number of elements, then they are isomorphic. We denote
by Cn a cyclic group with n elements (it is unique up to an isomorphism).
2. Normal subgroups and quotient groups
A subgroup H ⊆ G is said to be normal (notation: H C G) if gHg −1 ⊆ H for all
g ∈ G (it follows that gHg −1 = H for all g). The intersection of any family of normal
subgroups is also a normal subgroup. The kernel of any homomorphism is a normal
subgroup. If H C G, then G/H is endowed with a natural group structure. This
group is called the quotient group. The map π : G → G/H taking each element to
its coset is surjective and H = Ker π. The map π is called the canonical projection.
Many questions about a group G can be reduced to a normal subgroup H C G and
the quotient group G/H. A group G is called simple if it contains no nontrivial
normal subgroups.
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An element x ∈ G is called central if xy = yx for all y ∈ G. The set ZG of all
central elements is called the center of G. Let H ⊆ G be a subgroup. Define the
normalizer NH of H as {g ∈ G| gHg −1 ⊆ H}. This is the smallest subgroup of G,
in which H is normal. If S ⊆ G is an arbitrary subset, then define the centralizer
ZS of S as {g ∈ G | (∀x ∈ S) gx = xg}. This is the smallest subgroup of G s.t. S
lies in the center of it.
A tower G B G1 B · · · B Gr = ∅ is called a normal series. The number r
is the length of the normal series. If all the quotients Gi /Gi+1 are simple, then
the normal series is called compositional. Two normal series are called isomorphic if
their quotients are in a one-to-one correspondence such that corresponding quotients
are isomorphic.
Problems and theorems.
2.1. The homomorphism theorem. Let f : G → H be a homomorphism of groups. Then f (G)
is isomorphic to G/ Ker(f ).
2.2. Let F and H be subgroups of G. Prove the following statements:
• If F C H ⊆ G, then H ⊆ NF .
• If F ⊆ NH , then F H is a subgroup of G s.t. H C F H.
2.3. The isomorphism theorems. Prove the following statements:
• Let F C H C G. Then G/H = (G/F )/(H/F ) (here “=” means that there is a natural
isomorphism between the two groups).
• Let F and H be subgroups of G and H ⊆ NF . Then H/(H ∩ F ) = HF/F .
• Let f : G → G0 be a homomorphism and H 0 C G0 . Then H = f −1 (H 0 ) C G and
G/H = G0 /H 0 .
2.4. A normal series G0 B G1 B · · · B Gn−1 B Gn = ∅ is is said to be cyclic (resp, abelian) if all
the quotient groups Gi /Gi+1 are cyclic (resp., abelian). Prove that any abelian normal series can
be refined to a cyclic normal series.
2.5. Any normal series admits a refinement to a compositional series. Moreover, if normal series
G and H are isomorphic, then for any refinement of G there is an isomorphic refinement of H.
2.6. The Jordan-Hölder theorem. Any two normal series for G admit isomorphic refinements.
Any two compositional series are isomorphic. Hint: First find isomorphic refinements of G B G1 B
· · · B Gr = ∅ and G B H1 B ∅. Set D = G1 ∩ H1 , B = G1 H1 . Note that series B B G1 B D B ∅
and B B H1 B D B ∅ are isomorphic.
2.7. Any finitely generated abelian group is a quotient of Zn .
2.8. Let G be a group. The commutator of G is defined as the subgroup G0 of G generated by
{aba−1 b−1 | a, b ∈ G}. Prove that G0 C G, that G/G0 is abelian, and that for a normal subgroup
N of G, the quotient G/N is abelian iff G0 ⊆ N .
2.9. Suppose that N is a normal subgroup of G and G/N is cyclic. Then G is commutative.
3. Products of groups
Let G and H be groups. On the set G × H = {(g, h) | g ∈ G, h ∈ H}, introduce
the group structure as follows:
(g1 , h1 )(g2 , h2 ) = (g1 g2 , h1 h2 ).
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The group G × H is called the (direct) product of G and H. Note that G × H has
two distinguished normal subgroups G × {1} and {1} × H.
Problems and Theorems.
3.1. The groups G × H and H × G are isomorphic.
3.2. Let H and F be two subgroups of a group G. Assume that
• H ∩ F = {1},
• any element of H commutes with any element of F .
Then the subgroup HF ⊆ G is isomorphic to H × F .
3.3. The Chinese Remainder Theorem. Let p and q be relatively prime positive integers. The
product Cp × Cq is isomorphic to Cpq .
3.4. For a positive integer n, let ϕ(n) denote the number of positive integers from 1 to n that are
relatively prime to n. Show that the group Cn has ϕ(n) generators and that |Aut(Cn )| = ϕ(n).
The function ϕ is called the Euler ϕ-function.
3.5. Prove that ϕ(nm) = ϕ(n)ϕ(m) provided that m and n are relatively prime. If n = pa1 1 · · · pakk
is a prime decomposition of n, then
µ
¶ µ
¶
1
1
ak −1
a1 −1
ϕ(n) = p1 (p1 − 1) · · · pk
(pk − 1) = n 1 −
· 1−
.
p1
pk
3.6. Let N be a normal subgroup of a group G. Suppose that H is a subgroup of G such that
G = N H and N ∩ H = {1}. Then we say that G is a semidirect product of N and H. There exists
a homomorphism G → H which is the identity on H and whose kernel is N .
3.7. Let G be an abelian group. The torsion of G is defined as the set of all elements g ∈ G such
that ng = 0 for some n (we use additive notation). If G is finitely generated, then it is (isomorphic
to) a direct sum of its torsion and Z k .
4. Permutations
Let X be a finite set. A permutation of the set X is any one-to-one map from X to
itself. It follows that this map is also onto, just by counting elements. A permutation
of the set X = {x1 , . . . , xn } is also called a permutation of the elements x1 , . . . , xn .
Since the nature of the elements being permuted is not important, we can usually
set X = {1, 2, . . . , n}. The set of all permutations of 1, 2, . . . , n is denoted by Sn .
With any permutation of a finite set, we can associate its sign. To define the sign
of a permutation, it is convenient to number all elements, on which the permutation
acts. However, the sign will not depend on a particular numbering. Let σ be a
permutation of 1, 2, . . . , n. An inversion of σ is any pair (i, j), where 1 ≤ i < j ≤ n
but σ(i) > σ(j). The sign of σ is defined as the parity of the number of inversions.
I.e., the sign is 1 if the number of inversions is even and −1 if the number of inversions
is odd. A permutation is said to be even or odd depending on whether its sign is
equal to 1 or −1.
Let i and j be two numbers from 1 to n. Consider a permutation (ij) of the
numbers 1, 2, . . . , n, such that the image of i is j, the image of j is i and the image
of any other element, different from i and j, is this element itself. Such permutation
is called the transposition of i and j. A transposition is an odd permutation.
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Since permutations are maps, it makes sense to talk about the composition σ ◦τ of
two permutations σ and τ . This operation makes Sn a group, called a permutation
group, or a symmetric group.
Problems and theorems.
4.1. Any permutation is a composition of transpositions. Hint: It is always possible to reduce the
number of inversions by composing a given permutation σ with a suitable transposition. Namely,
there always exists an inversion (i, i+1), i.e. σ(i) > σ(i+1). Then the permutation (σ(i)σ(i+1))◦σ
has one inversion less than the permutation σ.
4.2. Composing a permutation with a transposition changes the sign of the permutation. In other
words, for any permutation σ and a transposition (ij), we have
sign(σ ◦ (ij)) = sign((ij) ◦ σ) = −sign(σ).
4.3. The sign of the composition of two permutations is the product of their signs.
4.4. A cycle (i1 . . . ik ) is a permutation of the set {1, 2, . . . , n} such that
σ(i1 ) = i2 ,
σ(i2 ) = i3 ,
...
σ(ik ) = i1
and σ(i) = i for any i different from all i1 , . . . , ik . In other terms, the elements i1 , . . . , ik are
permuted in the cyclic order. The number k is called the length of the cycle. The cycle of length
k has sign (−1)k−1 .
4.5. Two cycles (i1 . . . ik ) and (j1 . . . jl ) are called disjoint if the sets {i1 , . . . , ik } and {j1 , . . . , jl } are
disjoint. Any permutation can be represented as a product of disjoint cycles. This representation
is unique.
4.6. Suppose that σ ∈ Sn is represented as a product of disjoint cycles of lengths n1 , . . . , nk . The
order of σ is the least common multiple of n1 , . . . , nk .
4.7. Let An be the set of all even permutations in Sn . Then An is a normal subgroup of Sn of
index 2. The group An is called an alternating group.
4.8. The group An is generated by all cycles of length 3. Moreover, it is generated by all cycles of
the form (12k) for k = 3, . . . , n.
4.9. Describe all conjugacy classes in Sn . Answer: let n = n1 + · · · + nk be a partition of n
(the order of summunds is irrelevant). To each such partition, we can assign the conjugacy class
Sn [n1 , . . . , nk ] consisting of all permutations that decompose into independent cycles of lengths
n1 , . . . , nk . All conjugacy classes have this form.
4.10. Describe all conjugacy classes in An .
5. Group actions
Let X be an arbitrary set. A transformation of X is any map from X to itself
that is one-to-one and onto. All transformations of X form a group with respect to
the composition. In particular, if X is finite, this is a permutation group. An action
of a group G on a set X is any homomorphism from G to the transformation group
of X. For g ∈ G and x ∈ X, we sometimes write g(x) for the image of x under the
transformation corresponding to g. The set of all g ∈ G such that g(x) = x is called
the stabilizer of x. The set {g(x) | g ∈ G} is called the orbit of x.
Problems and theorems.
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5.1. Stabilizers of different elements in the same orbit are conjugate subgroups.
5.2. Let O be the orbit of x under an action of a group G. Denote by H the stabilizer of x. If G
is finite, then both H and O are finite. Moreover, |G| = |H| · |O|.
5.3. Let G be a finite group. Suppose that the index of each proper subgroup of G is divisible by
p. Then the center of G is non-trivial (i.e. has more than one element). Hint: consider the action
of G on itself by conjugations.
5.4. Let p > 0 be a prime. A finite group G is said to be a p-group if |G| = pn for some positive
integer n. If G is a p-group, then its center is non-trivial. Every p-group is solvable.
5.5. A subgroup H ⊆ G is called a Sylow’s p-subgroup if |H| is the maximal power of p that divides
|G|. Let G be a finite group and p a prime that divides |G|. Then G contains a Sylow’s p-subgroup.
Hint: use problem 3 and induction on |G|.
5.6. Sylow’s theorem. Let G be a finite group. Prove the following
• Any p-subgroup of G lies in a Sylow’s p-subgroup.
• All the Sylow’s p-subgroups are conjugate.
• The number of Sylow’s p-subgroups is congruent to 1 modulo p and divides the order of
G.
Hint: Let S be the set of all Sylow’s p-subgroups. Let G act on S by conjugations.
5.7. Let N be a normal subgroup of G and H another subgroup such that N H = G. Then H acts
on N by conjugations. If this action is trivial, and H ∩ N = {1}, then G is isomorphic to N × H.
5.8. Let G and H be two groups. Consider an action ϕ of H on G by automorphisms. Introduce
a group structure on G × H by the following rule:
(g1 , h1 )(g2 , h2 ) = (g1 ϕ(h1 )(g2 ), h1 h2 ).
Show that this is indeed a group structure. The group thus defined is called a semi-direct product
of G and H.
5.9. Let N be a normal subgroup of G and H another subgroup such that G = N H and N ∩ H =
{1}. Show that G is isomorphism to a semi-direct product of N and H.
5.10. The Classification theorem for finitely generated abelian groups. Any finitely
generated abelian group is a direct sum of cyclic groups. Moreover, one can arrange that any
summand is isomorphic either to Z or to Z/pk Z for a prime p and a positive integer k.
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