(Lec. 1; Artin ch. 2) A group is a set with an operation such that
there exists , the identity, such that for all ;
for all there exists an inverse such that ;
associativity: for all .
A group is abelian if is commutative ( for all ).
Examples:,,, the symmetric group ;, etc.; products,, …
(Lec. 2) Like sets, groups can be finite (,, …), countable (,,, …) or uncountable (, …).
is a subgroup if ;;. divides .
subgroups of is a subgroup of .
All subgroups of are for some .
A homomorphism is a map such that for all ().
Isomorphism = bijective homomorphism; automorphism = isomorphism . is a group.
The kernel of :, a subgroup of . injective .
The image of :, a subgroup of . surjective .
(Lec. 3) Given ,,, is a homomorphism with , the subgroup generated by .
, where = order of = .
Hence the cyclic group is if has order , and if has infinite order.
(generate if every element of is a product of the and their inverses.)
A subgroup determines an equivalence relation (axioms: ;; and ): (Lec. 4) iff , whose equivalence classes are the (left) cosets.
The quotient set: . The index of : if finite.
If is finite: subgroup divides ;; prime .
A subgroup is normal for all (left cosets = right cosets) for all (conjugate subgroups).
The operation makes a group iff is a normal subgroup.
For every homomorphism , is a normal subgroup of , and .
(Lec. 5) If is surjective, we have an exact sequence( injective, surjective, ).
Examples:( normal); ( iff );.
A homomorphism factors through iff .
is simple if its only normal subgroups are and itself. Examples:, prime; ,.
Example: the center is a normal subgroup (abelian: ).
Example: the commutator subgroup, where , is normal, and (abelianization) is the largest abelian quotient of : every with abelian factors as .
(Lec. 21; Artin 14.7) Every finitely generated abelian group is for some .
(Artin 6.7–6.12) Group actions: a -action on a set is ,, such that for all and ( a homomorphism ).
Faithful if is injective; transitive if for all there exists such that (i.e. one orbit).
(Lec. 22) The orbit of is . These form a partition orbits.
The stabilizer of is , a subgroup of .
Elements in the same orbit have conjugate stabilizer subgroups: .
Orbit-stabilizer: if , then ,, is a bijection; in particular .
Burnside’s lemma(, finite): let be the fixed points of ; then .
(Artin ch. 7) acts on itself by left multiplication. This gives , hence every finite group is isomorphic to a subgroup of ,.
acts on itself by conjugation: acts by .
(Lec. 24) Orbits = conjugacy classes; , the centralizer of .
Hence , where for each conjugacy class divides (the class equation of ).
For -groups (), the class equation (the number of conjugacy classes of size ). Hence:
, prime is abelian ( or ).
5 isomorphism classes of groups of order 8: ,,,, the quaternion group.
(Lec. 23) finite subgroup by considering the action of on its poles (unit vectors along rotation axes), one of , (regular -gon), (tetrahedron), (cube), (dodecahedron/icosahedron).
(Lec. 24) The symmetric group is generated by transpositions , in fact by .
For all there is a unique decomposition of as a product of disjoint cycles .
are in the same conjugacy class iff they have the same cycle lengths.
The alternating group.
(Lec. 25) A conjugacy class in which consists of even permutations is either 1 or 2 conjugacy classes in ; it splits into 2 iff the centralizer ( the cycle lengths of are all odd and distinct).
is simple for ( isn’t: is normal in and ).
(Lec. 26) Sylow theorems: if with , a Sylow -subgroup of is a subgroup of order .
Theorem 1: for every prime dividing , contains a Sylow -subgroup. ( consequence: contains an element of order .)
Theorem 2: all Sylow -subgroups of are conjugates of each other, and every subgroup of order () is contained in a Sylow subgroup.
Theorem 3: the number of Sylow -subgroups satisfies and .
If contains subgroups such that (e.g. because ) and , then for every there exist unique , such that .
If and are both normal in then .
If is normal but not , we have a semidirect product, with given by conjugation inside :.
(Lec. 28) Given (e.g. a -Sylow), the number of conjugate subgroups (e.g. all -Sylows) equals , where is the normalizer (the largest subgroup of such that is normal inside ).
Examples:
the Sylow subgroups of order 3 and 5 are normal ().
,, so either or the semidirect product .
1 or 3 2-Sylows, 1 or 4 3-Sylows; one of these is normal 5 isomorphism classes: ,,,,.
(Lec. 29) The free group (words in the never simplify except ).
Any group with generators is a quotient of , via ,.
is finitely presented if is generated by a finite set and their conjugates.
Write .
The Cayley graph of with generators : vertices = elements of ; edges: connect to for all , for all .
A normal form for elements of is a set of words in such that every element of appears exactly once among these.
(Lec. 30) Examples:.
is generated by ,..
(Lec. 30; Artin ch. 10; Fulton–Harris ch. 1–2) A representation of is a vector space on which acts by linear operators, i.e. a homomorphism .
A subrepresentation is a subspace invariant under : for all .
is irreducible if it has no nontrivial subrepresentations.
finite, finite-dimensional over : each has finite order, diagonalizable, with .
(Lec. 31) If is abelian, all operators are simultaneously diagonalizable irreducible representations are 1-dimensional.
These correspond to elements of the dual group. (Note is .)
A homomorphism of representations is a -equivariant linear map, i.e. .
representations of so are ,(),(),().( = the invariant part .)
Any -representation of a finite group admits an invariant Hermitian inner product, with respect to which acts by unitary operators.
a representation of a finite group (over ), an invariant subspace there exists invariant such that .
Hence any -representation of a finite group decomposes into a direct sum of irreducibles.
(Lec. 32) Schur’s lemma: irreducible representations of any homomorphism is either zero or an isomorphism; and all isomorphisms of an irreducible representation are multiples of :.
Example: representations of :
the trivial representation , acts by ;
the alternating representation , acts by ;
the standard representation (dimension ):, acts by permuting coordinates: .
,, are the only irreducible representations of .
(Lec. 33) The key tool to study representations is the character,.
(In terms of eigenvalues, and , so recovers all symmetric polynomial expressions in the , hence the as an unordered tuple.)
is a class function, i.e. .
,,,.
For a permutation representation ( acting on acts on with basis ,):.
(Lec. 34) Character table of = list, for each irreducible representation , of the value of on each conjugacy class.
is the projection onto , so .
is a Hermitian inner product on .
Then .
The characters of the irreducible representations of are an orthonormal basis of .
In particular, the number of irreducible representations = the number of conjugacy classes.
The multiplicities in the decomposition of a -representation into irreducibles, , are given by . Moreover, .
The regular representation of (= the permutation representation for acting on itself by left multiplication) contains each irreducible representation with multiplicity ; therefore .
(Lec. 35) These results allow us to find character tables of various groups (e.g. ,,,) by starting from known representations, considering tensor products, and using pairings and orthogonality to find irreducible pieces and the missing irreducible representations.