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.