1 SUBGROUPS

Basic structure theory

1 Subgroups

A subset H of elements of a group G is a subgroup, denoted H < G, if

the inverse of any element, and the product of any two elements of H

also belongs to H.

A subgroup is itself a group, and a subgroup of a subgroup is itself a

subgroup

K<H and H<G implies K<G

1 SUBGROUPS

Examples:

1. the set {1G} consisting of the identity element alone is a subgroup,

the trivial subgroup of G;

2. the additive group (2Z,+) of even integers is a subgroup of the

additive group (Z,+) of all integers;

3. the group U(1)={z∈C | |z|=1} of complex phases is a subgroup of

the multiplicative group C×=C\{0} of non-zero complex numbers;

4. the group of orientation preserving symmetries (i.e. rotations) of

a regular n-gon form a subgroup Cn of the dihedral group Dn;

1 SUBGROUPS

5. the centralizer CG(X) = {y∈G |xy=yx for all x∈G} of a subset

X ⊆ G, consisting of those group elements that commute with

all elements x ∈ X, is a subgroup of G, and in particular, the

center Z(G) = CG(G), consisting of those elements that commute

with every other element, is an Abelian subgroup (elements and

subgroups of the center are termed central);

6. the image φ(G)={φ(x) |x∈G} of a homomorphism φ :G→H is a

subgroup of the range, φ(G) < H;

7. the kernel kerφ={x∈G |φ(x)=1H} of a homomorphism φ :G→H

is a subgroup of the domain, kerφ < G;

8. the intersection of subgroups is again a subgroup.

1 SUBGROUPS

The set of subgroups form a complete lattice, i.e. a partially ordered set

in which any set S of subgroups has both an in�mum (the intersection⋂H∈S

H) and a supremum (the intersection of all subgroups containing

the subgroups in S), the subgroup lattice.

Subgroup lattice of D3

Si = {1, σi} = 〈σi〉 ∼= Z2

C3 ={1, C, C-1

}= 〈C〉 ∼= Z3

1 SUBGROUPS

For any subset X ⊆ G of group elements, the closure

〈X〉 =⋂

X⊆H<G

H

(aka. the subgroup generated by X) is the least subgroup of G that

contains all elements of X (it contains all possible products of elements

of X and of their inverses).

A subset X ⊆ G generates a subgroup H < G if 〈X〉=H; if 〈X〉=G,

then X is a system of generators for G.

A group (subgroup) is �nitely generated if it has a �nite generating

system, and cyclic if it can be generated by a single element.

2 CYCLIC SUBGROUPS

2. Cyclic subgroups

Powers of a group element x∈G de�ned recursively: xn+1=xxn.

. . . , x-3=x-1x-2, x-2=x-1x-1, x-1, x0=1G, x1=x, x2=xx, x3=xx2, . . .

Multiplication of powers! addition of exponents xnxm = xn+m

Corollary. The mapφx :Z→ G

n 7→ xn

is a homomorphism for any x∈G, i.e. φx(n+m)=φx(n)φx(m).

Since the (cyclic) subgroup⟨x⟩generated by x contains all powers of x,

φx(Z)={xn |n∈Z}=⟨x⟩

2 CYCLIC SUBGROUPS

If H <G is cyclic, i.e. H =⟨x⟩for some x∈G (the cyclic generator of

H), then either

· all powers of x di�er, i.e. xn 6= xm if n 6=m, in which case φx is

bijective, and H is isomorphic to the additive group of integers,

H∼=Z (in�nite cyclic case), or

· some power of x is the identity (the smallest positive integer N

such that xN =1G is the order of x), implying that φx is constant

on residue classes modulo N H is isomorphic with the (�nite)

additive group ZN of residue classes mod N (�nite cyclic case).

2 CYCLIC SUBGROUPS

Order of a cyclic group = order of its cyclic generator.

Addition is commutative cyclic groups are Abelian!

Finitely generated Abelian groups can be obtained from cyclic groups.

Structure of cyclic groups

1. the order of a cyclic group is either �nite or countably in�nite;

2. two cyclic groups are isomorphic precisely when they have the same

order.

3 COSETS

3 Cosets

A coset of a subgroup H < G is a set of group elements of the form

xH = {xh |h∈H} left

Hx = {hx |h∈H} right

for some x∈G (with 1H=H1=H the trivial coset).

Unless G is Abelian, left and right cosets usually di�er, xH 6=Hx.

The subgroup H < G is normal, denoted H/G, if all left and right cosets

coincide, i.e. xH=Hx for all x∈G.

The coset spaces G/H = {xH |x∈G} and H\G= {Hx |x∈G} are the

collections of left and right cosets of H.

3 COSETS

Examples:

1. the cosets (left or right) of the trivial subgroup {1G} < G are the

one element sets {x} for x∈G;

2. (2Z,+) < (Z,+) has two cosets (even and the odd numbers);

3. the coset space of C×/U(1) is in one-to-one correspondence with

the positive real numbers;

4. the rotation subgroup Cn /Dn has two cosets, the trivial coset

consisting of orientation preserving symmetries (rotations), and

the coset σ1Cn = {σ1, . . . , σn} consisting of orientation reversing

symmetries (re�ections).

3 COSETS

There is a bijective correspondence between G/H and H\G, hence it is

enough to study left cosets.

The index [G : H] of a subgroup H < G is the cardinality of its coset

space, i.e.

[G : H] = |G/H| = |H\G|

Poincaré's theorem: the intersection of two �nite index subgroups is also

of �nite index.

Corollary. Finite index subgroups form a sublattice of the full subgroup

lattice.

3 COSETS

Cosets equipartition the group, that is, each group element belongs to

exactly one coset, and each coset has the same cardinality (equal to the

order of H).

Consequences:

1. the relation x ≡H y if x-1y ∈H is an equivalence relation, whose

equivalence classes are precisely the cosets of H;

2. for subgroups H<G of a �nite group

|G|=[G :H]|H| Lagrange'stheorem

the order of any subgroup divides the order of the group;

3. groups of prime order are cyclic (since 〈x〉 = G for 1G 6= x∈G).

4 NORMAL SUBGROUPS

4 Normal subgroups

Normal subgroup: N / G if xN=Nx for all x∈G.

The trivial subgroup the whole group are always normal: a group is

simple, if it has no other normal subgroup.

All subgroups of an Abelian group are normal a �nite Abelian group

is simple if it is cyclic of prime order.

Simple groups may be viewed as the elementary (atomic) constituents of

which more general groups may be constructed many general theorems

may be reduced to the case of simple groups.

4 NORMAL SUBGROUPS

Finite simple groups:

1. the cyclic groups Zp of prime order;

2. the alternating groups An for n > 4;

3. the �nite Lie groups (�nite analogs of Lie groups, falling into several

in�nite families, together with suitable twisted versions);

4. 26 sporadic groups (including the famous Mathieu groups as well

as the Monster M, with more than 1058 group elements).

Intersection of normal subgroups is normal normal subgroups form a

(modular) sublattice of the subgroup lattice.

4 NORMAL SUBGROUPS

Congruence relation: equivalence relation compatible with binary oper-

ation.

x1 ≡ y1

x2 ≡ y2

⇒ x1x2 ≡ y1y2

One-to-one correspondence between normal subgroups and congruence

relations, given by

· cosets of a normal subgroup are the equivalence classes of a con-

gruence relation;

· the congruence class {x∈G |x ≡ 1G} of the identity element is a

normal subgroup.

5 FACTOR GROUPS

5 Factor groups

Extension of group product to subsets X,Y ⊆G of group elements

XY = {xy |x∈X, y∈Y } ⊆ G

Associative operation on subsets, but inverses exist only for singletons.

The product of cosets is usually the union of several cosets, but for a

normal subgroup, the product of cosets is a single coset!

Consequence: the cosets of a normal subgroup form a group (with

the trivial coset as identity element), the factor group G/N , of order

|G/N | = [G :N ].

5 FACTOR GROUPS

Examples:

1. the factor group G/ {1G} is isomorphic to G;

2. C×/U(1) is isomorphic to the multiplicative group of positive real

numbers;

3. the factor group Dn/Cn has order 2, hence isomorphic to Z2;

4. An is a normal subgroup of index 2 in Sn, hence Sn/An∼= Z2.

Remark. In general, any subgroup of index 2 is normal, and the corre-

sponding factor group is isomorphic to Z2.

5 FACTOR GROUPS

(xN)(yN) = (xy)N

for x, y∈G and N/G, hence the map

$N :G→ G/N

x 7→ xN

is a surjective homomorphism, the natural homomorphism.

First isomorphism theorem: if N/ G and H<G, then N/ NH<G

and N∩H/H; moreover

H/(N∩H)∼=NH/N

Second isomorphism theorem: if K,N/ G then N/K/ G/K, and

(G/K)/(N/K)∼=G/N

6 THE HOMOMORPHISM THEOREM

6 The homomorphism theorem

The kernel of a homomorphism φ :G→H is a normal subgroup of the

domain G, kerφ / G, and the factor group G/ kerφ is isomorphic with

the image φ(G): G/ kerφ ∼= φ(G).

The homomorphic images of a given group are, up to isomorphism, pre-

cisely its the factor groups.

Correspondence theorem: one-to-one correspondence between sub-

groups of the image and those subgroups of the domain that contain the

kernel.

{H < G | kerφ < H}! {K < φ(G)}

7 FREE GROUPS

7 Free groups

A group F is free if it has a free generating system X ⊆F , i.e. one for

which every map φ : X → G into some group G is the restriction of a

unique homomorphism φ[ :F→G.

For any set X there exists a group FX , called the free group over X,

with free generating set X, and FX∼= FY exactly when

∣∣X∣∣= ∣∣Y ∣∣ .Corollary. To each cardinal number corresponds exactly one isomor-

phism class of free groups, and any two free generating systems of a free

group F have the same cardinality rk(F ), called its rank.

7 FREE GROUPS

Any group is the homomorphic image of a free group!

Proof : If the group G is generated by X⊆G, then the inclusion map

iX :X → G

x 7→ x

extends to a unique surjective homomorphism i[X :FX→G. �

According to the above, any group G can be presented as a factor group

FX/NX

for any choice of generating set X, where NX =ker i[X .

Question: can we characterize NX e�ectively?

7 FREE GROUPS

Nielsen-Schreier theorem: any subgroup of a free group is itself free.

NX can be characterized by specifying a free generating set (ine�ective

for in�nite groups, in which case ker i[X has in�nite rank).

Normal closure: smallest normal subgroup (intersection of all normal

subgroups) containing a given set of group elements.

Given a generating system X ⊆G, a set of relators is a subset R⊆FX

whose normal closure is ker i[X .

A presentation 〈X|R〉 of G consists of a generating set X and a corre-

sponding set of relators. If both X and R are �nite, then G is �nitely

presented (amenable to algorithmic methods).

8 DIRECT PRODUCT OF GROUPS

8 Direct product of groups

The direct product G1×G2 of the groups G1 and G2 has as elements

the ordered pairs (x1, x2) with x1∈G1 and x2∈G2, and component-wise

multiplication

(x1, x2) (y1, y2) = (x1y1, x2y2)

G1×G2 is a group of order |G1×G2|= |G1| |G2|, with inverses given by

(x1, x2)-1=(x-11 , x

-1

2

).

The direct product is commutative and associative (up to isomorphism)

G1×G2∼=G2×G1 and G1×(G2×G3)∼=(G1×G2)×G3

8 DIRECT PRODUCT OF GROUPS

Examples:

1. if (n,m) denotes the greatest common divisor, and [n,m] the least

common multiple of the integers n and m, then

Zn × Zm∼= Z[n,m] × Z(n,m)

and in particular Zn×Zm∼= Znm for coprime n and m;

2. U(n)∼=SU(n)×U(1);

3. D4n+2∼= D2n+1×Z2.

8 DIRECT PRODUCT OF GROUPS

Natural projections (surjective homomorphisms)

πi :G1×G2 → Gi

(x1, x2) 7→ xi

onto the factors.

Structural characterization:

G1 = ker(π2) = {(x1, 1) |x1∈G1} and G2 = ker(π1) = {(1, x2) |x2∈G2}

are normal subgroups of the direct product, such that

1. they have trivial intersection, G1 ∩ G2={(1, 1)};

2. their elements commute pairwise, (x1, 1)(1, x2)=(x1, x2)=(1, x2)(x1, 1);

3. they generate the whole product, G1G2 = G1×G2.

8 DIRECT PRODUCT OF GROUPS

Conversely: any group having two normal subgroups satisfying the above

is isomorphic to their direct product.

Remark. Gi∼= Gi and (G1×G2) /Gi

∼= G3−i for i=1, 2.

Many commuting elements in a direct product mostly useful for com-

mutative groups.

Direct product of Abelian (in particular cyclic) groups is an Abelian

group.

Frobenius-Stickelberger theorem: any �nite Abelian group can be

decomposed, uniquely up to ordering, into the direct product of cyclic

groups of prime power order; in the �nitely generated case, in�nite cyclic

factors may also appear.

9 SOLUBLE GROUPS

9 Soluble groups

The commutator of the group elements x, y∈G is

[x, y]=x-1y-1xy

[x, y]=1 only in case xy=yx, i.e. if x and y commute.

The commutator (or derived) subgroup G′ is the subgroup generated by

all commutators

G′=〈{[x, y] |x, y∈G}〉

G′ is trivial only when G is Abelian; the group G is perfect if G′=G (a

simple group is either commutative or perfect).

9 SOLUBLE GROUPS

G′ / G, and the factor group G/G′ is Abelian (what is more, G′ is the

smallest normal subgroup N / G such that G/N is commutative).

The derived series

G0=G . G1=G′0 . G2=G

′1 . · · ·

is subnormal in the sense that each term Gi is normal in the previous

term, Gi/Gi−1, being the derived subgroup of Gi−1.

A group G is soluble, if Gn = {1} after �nitely many steps n (relaxed

commutativity).

Feit-Thompson theorem: groups of odd order are soluble.

9 SOLUBLE GROUPS

Importance of soluble groups: Galois theory, di�erential equations, gauge

theories, algorithmic methods, etc.

Composition series: a subnormal series

G=G0 . G1 . · · · . Gn={1}

where all composition factors Gi−1/Gi are simple groups.

Remark. Not all groups have a composition series (but many, e.g. soluble

ones, have).

Jordan-Hölder theorem: if a group has a composition series, then all

its composition series have equal length, and their composition factors

coincide (up to order).

10 CONJUGACY CLASSES

10 Conjugacy classes

For each z∈G, the map

cz :G→ G

x 7→ zxz-1

is an automorphism cz∈Aut(G), called an inner automorphism.

Proof. cz(xy)=z(xy)z-1=(zxz-1)(zyz-1)=cz(x) cz(y)

Notice that cy(x)=x when x and y commute (and then cx(y)=y), i.e.

CG(x) = {y∈G | cy(x)=x}

and in particular cy= idG precisely when y∈Z(G).

10 CONJUGACY CLASSES

The map

Inn :G→ Aut(G)

z 7→ cz

is a homomorphism, whose kernel equals the center Z(G) of G.

Proof : (cx◦cy)(z)=cx(yzy-1

)=x(yzy-1

)x-1=(xy)z(xy)

-1=cxy(z) �

The image Inn(G), isomorphic to G/Z(G) by the homomorphism theo-

rem, is normal in Aut(G), and the factor group

Out(G) = Aut(G) /Inn(G)

is called the group of outer automorphisms of G.

10 CONJUGACY CLASSES

The conjugacy class xG (resp. HG) of a group element x ∈ G (resp.

subgroup H<G) consist of the image of x (resp. H) under all inner

automorphisms

xG = {cy(x) | y∈G}

HG = {cy(H) | y∈G}

Conjugacy classes of elements (resp. subgroups) partition the group

(resp. subgroup lattice): conjugacy classes are either disjoint or equal.

A group element is central (a subgroup is normal) if it is the only ele-

ment of its conjugacy class conjugacy classes of Abelian groups are

singletons (one element sets).

10 CONJUGACY CLASSES

The cardinality of conjugacy classes is given by∣∣xG∣∣ =[G : CG(x)]∣∣HG∣∣ = [G : NG(H)]

where NG(H)={x∈G |xH=Hx} is the normalizer of H<G, i.e. the

smallest subgroup of G in which H is normal.

Example: The conjugacy classes of D3 are

C1={1} , C2={C,C-1

}and C3={σ1, σ2, σ3}

The subgroups of D3 form 4 conjugacy classes: since the trivial subgroup

{1}, the rotation subgroup C3={1, C, C-1

}and D3 itself are all normal,

they each form a conjugacy class in themselves, whereas the subgroups

〈σi〉={1, σi} generated by the re�ections form a single conjugacy class.