Fundamental concepts
What’s a homomorphism? What are its properties?
It’s a map from one group to another that is structure preserving (the image of a product is the product of the images).
- isomorphisms are bijective homomorphisms
- they preserve the identity elements and commute with taking inverses
- they can be composed
- image of a homomorphism (G —» H): the subset of H that consists of the images of the elements of G
- kernel: a subset of G, the collection of elements mapped onto the identity element of H
- they are surjective when the image of G is H
- they are injective when the only element of the kernel is the identity element of G
Subgroups
What are some examples of subgroups?
- trivial subgroup: only element is the identity
- additive group of even integers < additive group of all integers
- image of a homomorphism, ϕ(G) < range of the homomorphism, H
- kernel of a homomorphism, kerϕ < domain of the homomorphism, G
Subgroups
What’s the subgroup lattice and how can it be visualized?
The intersection of subgroups is again a subgroup, hence the subgroups of G form the subgroup lattice.
- any collection S of subgroups has both a greatest lower bound (their intersection) and a lowest upper bound (the intersection of all subgroups containing the elements of S)
- visualization: Hasse-diagram (vertices correspond to different subgroups, two vertices are connected if theirs a subgroup ordering between the two)
Subgroups
What’s Cayley’s theorem? What are its implications?
Every finite group is isomorphic to a subgroup of some symmetric group of finite degree.
- from the fact that every subgroup of a symmetric group of finite degree is finite
- implication: the study of finite groups could be reduced to that of groups of permutations (but this is not always convenient)
Generating sets
What is a generating set? Properties? Their use?
G(cursive) is a generating set (aka. system of generators) for H – if H is the smallest subgroup of G containing G(cursive) (i.e. H is the intersection of all subgroups of G containing G(cursive)).
- a subgroup generated by the generating set contains all possible products of elements of the generating set, along with their inverses
- finitely generated group: it has a finite generating set
- cyclic group: it is generated by a single element
- a group (subgroup) may have many different generating sets, since any set of group elements that contain a generating set is itself a generating set
- use: group description (specification of a particular group as a subgroup of some bigger, well understood group via a generating set)
Cosets
What are cosets? Some examples? Why is it enough to study left cosets?
A coset of a subgroup H < G is a set of group elements of the form
left: xH = {xh|h ∈ H},
right: Hx = {hx|h ∈ H}
for some x ∈ G.
- trivial cosets: 1H = H1 = H
- left and rigth cosets usually differ
- coset spaces: the collections of left and right cosets (G/H = {xH | x ∈ G}, H\G = {Hx | x ∈ G})
- there is a bijective correspondence between G/H and H\G, hence it is enough to study left cosets
Examples:
- (2Z, +) has two cosets (even, odd numbers)
- the coset space of C×/U(1) is in one-to-one correspondence with the positive real numbers
- rotation subgroup has two cosets: reflections (orientation non-preserving), rotations (orientation preserving)
Cosets
What’s Poincaré’s theorem?
The intersection of two finite index subgroups is again of finite index.
- index [G:H] of a subgroup H < G: cradinality of its coset spaces
- the cosets of a subgroup equipartition the set of group elements: each group element belongs to exactly one coset, and each coset has the same cardinality (equal to that of the trivial coset, i.e. the order of H)
Cosets
What is Lagrange’s theorem?
If G is finite and H < G, then |G| = [G : H]|H|. In particular, the order of any subgroup divides the order of the group.
- corollary: groups of prime order are cyclic
- not only the order, but the index of a subgroup is also a divisor of the order of the whole group
Normal subgroups
What’s a normal subgroup? What are its properties?
A subgroup N < G is a normal subgroup, denoted N ◁ G, if its right cosets coincide with its left cosets, i.e. xN = N x for all x ∈ G.
- the trivial subgroup and the whole group are always normal
- simple subgroup: a group which has no other normal subgroup apart from the trivial subgroup
- all subgroups of an Abelian group are normal (a finite Abelian group is simple if and only if it is cyclic of prime order)
- the intersection of normal subgroups is normal again
- one-to-one correspondence between normal subgroups and congruence relations
Factor groups
What are factor groups? Examples?
Collection of cosets of the normal subgroup N ◁ G with the product below (with the trivial coset as identity element):
(xN )(yN ) = (xy)N,
for a normal subgroup N ◁ G and group elements x, y ∈ G.
- for a normal subgroup, the product of cosets is a single coset
- if p denotes the smallest prime divisor of the order, than any subgroup of index p is normal, and the corresponding factor
group is isomorphic with Z(p)
Examples:
- the factor group G/{1G} is isomorphic to G
- the factor group D(n)/C(n) is isomorphic to Z2
Factor groups
What s the correspondence theorem?
Subgroups of the factor group G/N are of the form H/N , where H < G is a subgroup of G containing the normal subgroup N (with normal subgroups corresponding to normal ones).
- in other words, the (normal) subgroup lattice of the factor group G/N is completely determined by the (normal) subgroup lattice of G
Factor groups
What are the isomorphism theorems?
- If N ◁ G and H < G, then:
- N ◁ NH < G
- N ∩ H ◁ H
- H/(N∩H) ∼= NH/N
- If K ◁N ◁ G and K ◁G, then:
- N/K ◁ G/K
- (G/K)/(N/K) ∼= G/N
Subnormal series and soluble groups
What are subnormal series and composition series?
Subnormal series: a finite sequence of subgroups G = G0 ▷ G1 ▷ · · · ▷ Gn = {1}, where each term is a normal subgroup of the preceding one
Composition series: a subnormal series where all factor groups G(i−1)/G(i) (the composition factors) are simple groups
- these provide the dissection of group “molecules” into their “atomic constituents”
Subnormal series and soluble groups
What’s the Jordan-Hölder theorem?
If a group has several composition series, then all have the same length, and their composition factors coincide.
- all finite groups have a (or possibly several) composition series, but infinite ones don’t necessarily
Subnormal series and soluble groups
When is a group soluble? What’s the Feit-Thompson theorem?
A group is soluble if it has a subnormal series where all factor groups G(i−1)/G(i) are Abelian (commutative).
- application: differential equations, Galois theory, etc.
- solubility is a kind of relaxed commutativity
- all subgroups and factor groups of a soluble group are themselves soluble
- example: S(n) is soluble only for n =< 4 (equations above degree 4 cannot be solved)
Feit-Thompson theorem: finite groups of odd order are soluble
Subnormal series and soluble groups
What’s the commutator of group elements? What’s a derived series?
The commutator of group elements x,y ∈ G is the group element [x,y] = x^(-1)y^(-1)xy.
- commutator (or derived) subgroup: the subgroup G′ = ⟨{[x, y]|x, y ∈ G}⟩ generated by all commutators
- two group elements commute if their commutator equals the identity, so G’ is trivial only when G is Abelian
- G’ is always a normal subgroup of G and G/G’ is always Abelian
- derived series: G = G0 ▷ G1 ▷ · · · ▷ Gi, with G(i+1) = G′(i) (need not terminate in the trivial subgroup)
- group G is soluble if its derived series is subnormal, i.e. reaches the trivial subgroup in a finite number of steps
Homomorphism theorem
What’s a natural projection? What’s the homomorphism theorem?
For a normal subgroup N ◁ G, the natural projection: πN : G → G/N, x → xN is a homomorphism with kernel equal to N.
Homomorphism theorem: the kernel of a homomorphism ϕ : G → H is a normal subgroup, ker ϕ ◁ G, and its image is isomorphic with the corresponding factor group, ϕ(G) ∼= G/ker ϕ.
- the homomorphic images (an external attribute) of a given group coincide with its factor groups (an internal attribute)
Cyclic (sub)groups
What are cyclic subgroups?
The smallest subgroup containing x ∈ G (the group ⟨x⟩ generated by it) has as elements its different powers: ⟨x⟩ = {x^n | n ∈ Z}.
- powers of the group element x ∈ G are defined recursively as x^1 = x and x^(n+1) = xx^n for an integer n
- any cyclic group is Abelian because the addition of integers is commutative
- the map ϕ(x) : Z → G that assigns to each integer n the nth power of x ∈ G, i.e. ϕ(x)(n) = x^n, is actually a homomorphism
ϕ(x)(n+m) = x^(n+m) = x^n x^m = ϕ(x)(n)ϕ(x)(m) - the image of ϕ(x) consists of all powers of x, hence x = ϕ(x)(Z) ∼= Z/ kerϕ(x)
Cyclic (sub)groups
What is the structure theorem of cyclic groups?
The order of a cyclic group is either finite or countably infinite, and two cyclic groups are isomorphic precisely when they have the same order.
Direct product of groups
What is the direct product of groups? Some examples?
The direct product G×H of the groups G and H is a new group, whose elements are ordered pairs (x, y) with x ∈ G and y ∈ H, endowed with component-wise multiplication:
(x1, y1) (x2, y2) = (x1x2, y1y2).
- G × H has order |G×H| = |G||H|, its identity element is (1G, 1H ), and inverses are given by (x, y)^(-1) = x^(-1), y^(-1))
- it’s a binary operation between isomorphism classes of groups (so it’s commutative and associative)
Examples:
- GL(n)(C) ∼= SL(n)(C) × C×
- the additive group (V, +) of a linear space of dimension n over a field F is isomorphic with the n-fold direct product of the additive group of F
Direct product of groups
When are groups isomorphic to a direct product?
G^ = {(x, 1H ) | x ∈ G} and H^ = {(1G, y) | y ∈ H} are normal subgroups of the direct product that
- generate the whole product, ⟨G^, H^⟩ = G×H,
- have trivial intersection, G^ ∩ H^ = {(1G, 1H )},
- have pairwise commuting elements: (x, 1H )(1G, y) = (x, y) = (1G, y)(x, 1H ).
If G^ and H^ satisfy 1)-3), then any group that has them as normal subgroups is isomorphic to G^ × H^.
Direct product of groups
What’s the Frobenius-Stickelberg theorem?
Any finite Abelian group can be decomposed into a direct product of cyclic groups of prime power order.
- consequence of the fact that the direct product of Abelian (in particular cyclic) groups is Abelian
- for finitely generated Abelian groups: a finite number of infinite cyclic factors may also appear in the decomposition
Group presentations
What are free generating systems and free groups? Two theorems?
A subset X ⊆ F is a free generating system of the group F if every map ϕ : X → G into an arbitrary group G is the restriction of a unique homomorphism ϕ♭ : F → G.
A group is free if it has a free generating system.
- for any set X there exists a group FX (the free group over X) with free generating set X
- any two free generating systems
of a free group have the same cardinality
Nielsen-Schreier theorem: every subgroup of a free group is free
von Dyck’s theorem: every group is a homomorphic image of a free group
Group presentations
What’s a presentation? What’s the basic algorithm problem?
A presentation ⟨X|R⟩ of the group G consists of a generating set X ⊆ G and a subset R ⊆ FX (relators) whose normal closure is the kernel of i♭(X) .
- inclusion map: i(X) : X → G that sends each x ∈ X to itself extends to a unique homomorphism i♭(X) : FX → G
- normal closure of a group: the intersection of all normal subgroups containing it
Basic algorithmic problem (word problem): for a finite presentation ⟨X|R⟩, decide whether two elements w1, w2 ∈ FX are mapped to the same element, i.e. whether i♭(X)(w1) = i♭(X)(w2)
