(MLA) F (vσ(1) , . . . , vσ(k) , . . . , vσ(r) ) = sign(σ)F (v1 , . . . , vk , . . . , vr ),
where sign(σ) ∈ {±1} is the sign of σ.
The tensor product of V1 , . . . , Vr is a kvector space V1 —V2 —· · ·—Vr together with a function
„ : V1 — · · · — Vr ’’ V1 — V2 — · · · — Vr satisfying the following universal property.
UPTP: For any kvector space W and multilinear map F : V1 — · · · — Vr ’’ W , there is a
unique linear transformation F : V1 — · · · — Vr ’’ W for which F —¦ „ = F .
10 1. LINEAR AND MULTILINEAR ALGEBRA
In diagram form this becomes
„ / V1 — · · · — Vr
V1 — · · · — Vr
LLLL
LLL
LLL
F ∃! F
L& x
W
where the dotted arrow represents a unique linear transformation making the diagram commute.
When V1 = V2 = · · · = Vr = V , we call V — · · · — V the rth tensor power and write Tr V .
The following result provides an explicit description of a tensor product.
Proposition 1.15. If the ¬nite dimensional kvector space Vk (1 k r) has a basis
vk = {vk,1 , . . . , vk,nk }
where dim Vk = nk , then V1 — · · · — Vr has a basis consisting of the vectors
v1,i1 — · · · — vr,ir = „ (v1,i1 , . . . , vr,ir ),
where 1 ik nk . Hence we have
dim V1 — · · · — Vr = n1 · · · nr .
More generally, for any sequence w1 ∈ V1 , . . . wr ∈ Vr , we set
w1 — · · · — wr = „ (w1 , . . . , wr ).
These satisfy the multilinearity formul¦
(MLF1) w1 — · · · — wk’1 — (wk + wk ) — wk+1 — · · · — wr =
w1 — · · · — wk — · · · — wr + w1 — · · · — wk’1 — wk — wk+1 — · · · — wr ,
(MLF2)
w1 — · · · — wk’1 — twk — wk+1 — · · · — wr = t(w1 — · · · — wk’1 — wk — wk+1 — · · · — wr ).
We will see later that the tensor power Tr V can be decomposed as a direct sum Tr V =
Symr V • Altr V consisting of the symmetric and antisymmetric or alternating tensors Symr V
and Altr V .
We end with some useful results.
Proposition 1.16. Let V1 , . . . , Vr be ¬nite dimensional kvector spaces. Then there is a
linear isomorphism
V1— — · · · — Vr— ∼ (V1 — · · · — Vr )— .
=
In particular,
Tr (V — ) ∼ (Tr V )— .
=
Proof. Use the universal property to construct a linear transformation with suitable prop
erties.
Proposition 1.17. Let V, W be ¬nite dimensional kvector spaces. Then there is a klinear
isomorphism
W — V — ∼ Hom (V, W )
=
under which for ± ∈ V — and w ∈ W ,
w — ± ←’ w±
where by de¬nition, w± : V ’’ W is the function determined by w±(v) = ±(v)w for v ∈ V .
4. BASIC NOTIONS OF MULTILINEAR ALGEBRA 11
Proof. The function W — V — ’’ Hom (V, W ) given by (w, ±) ’ w± is bilinear, and
hence factors uniquely through a linear transformation W — V — ’’ Hom (V, W ). But for bases
—
v = {v1 , . . . , vm } and w = {w1 , . . . , wn } of V and W , then the vectors wj — vi form a basis of
W — V — . Under the above linear mapping, wj — vi gets sent to the function wj vi which maps
— —
vk to wj if k = i and 0 otherwise. Using Propositions 1.2 and 1.15, it is now straightforward to
verify that these functions are linearly independent and span Hom (V, W ).
Proposition 1.18. Let V1 , . . . , Vr , W1 , . . . , Wr be ¬nite dimensional kvector spaces, and
for each 1 k r, let •k : Vk ’’ Wk be a linear transformation. Then there is a unique linear
transformation
•1 — · · · — •r : V1 — · · · — Vr ’’ W1 — · · · — Wr
given on each tensor v1 — · · · — vr by the formula
•1 — · · · — •r (v1 — · · · — vr ) = •1 (v1 ) — · · · — •1 (vr ).
Proof. This follows from the universal property UPTP.
CHAPTER 2
Recollections and reformulations on basic group theory
Recommended Books.
J&L: G. James & M. Liebeck, Representations and Characters of Groups, Cambridge Uni
versity Press, 1993.
JPS: J.P. Serre, Linear Representations of Finite Groups, SpringerVerlag, New York, 1977.
A&B: J.L. Alperin & R.B. Bell, Groups and Representations, SpringerVerlag, New York,
1995.
JBF: J.B. Fraleigh, A First Course in Abstract Algebra, AddisonWesley, 1994 (5th Edition).
1. The Isomorphism and Correspondence Theorems
The three Isomorphism Theorems and the Correspondence Theorem are fundamental results
of Group Theory. We will use the notations H G and N G to indicate that H is a subgroup
and N a normal subgroup of G.
Recall that given a normal subgroup N G the quotient group G/N has for its elements the
distinct cosets
gN = {gn ∈ G : n ∈ N } (g ∈ G).
Then the natural mapping π : G ’’ G/N given by
π(g) = gN
is a surjective homomorphism with kernel ker π = N .
Theorem 2.1 (1st Isomorphism Theorem). Let • : G ’’ H be a homomorphism with
N = ker •. Then there is a unique homomorphism • : G/N ’’ H such that • —¦ π = •.
Equivalently, there is a unique factorisation
•
π
• : G ’ G/N ’ H.
’ ’
In diagram form this becomes
q
/ G/N
G?
??
??
?
•? ? ∃! •

H
where all the arrows represent group homomorphisms.
Theorem 2.2 (2nd Isomorphism Theorem). Let H G and N G. Then there is an
isomorphism
HN/N ∼ H/(H © N ); hn ←’ h(H © N ).
=
Theorem 2.3 (3rd Isomorphism Theorem). Let K G and N G with N K. Then
K/N G/N is a normal subgroup, and there is an isomorphism
G/K ∼ (G/N )/(K/N ); gK ←’ (gN )(K/N ).
=
13
14 2. RECOLLECTIONS AND REFORMULATIONS ON BASIC GROUP THEORY
Theorem 2.4 (Correspondence Theorem). There is a oneone correspondence between sub
groups of G containing N and subgroups of G/N , given by
H ←’ π(H) = H/N,
π ’1 Q ←’ Q,
where
π ’1 Q = {g ∈ G : π(g) = gN ∈ Q}.
Moreover, under this correspondence, H G if and only if π(H) G/N .
2. Some de¬nitions and notation
Let G be a group.
Definition 2.5. The centre of G is the subset
Z(G) = {c ∈ G : gc = cg ∀g ∈ G}.
This is a normal subgroup of G, i.e., Z(G) G.
Definition 2.6. Let g ∈ G, then the centralizer of g is
CG (g) = {c ∈ G : cg = gc}.
This is a subgroup of G, i.e., CG (g) G.
Definition 2.7. Let H G. The normalizer of H in G is
NG (H) = {c ∈ G : cHc’1 = H}.
This is a subgroup of G containing H; moreover, H is a normal subgroup of NG (H), i.e.,
H NG (H).
Definition 2.8. G is simple if its only normal subgroups are G and {e}. Equivalently, it
has no nontrivial proper subgroups.
Definition 2.9. The order of G, G, is the number of elements in G when G is ¬nite, and
∞ otherwise. If g ∈ G, the order of g, g, is the smallest natural number n ∈ N such g n = e
provided such a number exists, otherwise it is ∞. Equivalently, g =  g , the order of the
cyclic group generated by g. If G is ¬nite, then every element has ¬nite order.
Theorem 2.10 (Lagrange™s Theorem). If G is a ¬nite group, and H G, then H divides
G. In particular, for any g ∈ G, g divides G.
Definition 2.11. Two elements x, y ∈ G are conjugate in G if there exists g ∈ G such that
y = gxg ’1 .
The conjugacy class of x is the set of all elements of G conjugate to x,
{y ∈ G : y = gxg ’1 for some g ∈ G}.
Conjugacy is an equivalence relation on G and the distinct conjugacy classes are the distinct
equivalence classes.
3. GROUP ACTIONS 15
3. Group actions
Let G be a group (with identity element eG ) and X a set. Recall that an action of G on X
is a rule assigning to each g ∈ G a bijection •g : X ’’ X and satisfying the identities
•gh = •g —¦ •h ,
•eG = IdX .
We will frequently make use of the notation
g · x = •g (x)
(or even just write gx) when the action is clear, but sometimes we may need to refer explicitly
to the action. It is often useful to view an action as corresponding to a function
¦ : G — X ’’ X; •(g, x) = •g (x).
It is also frequently important to regard an action of G as corresponding to a group homomor
phism
• : G ’’ Perm(X); g ’ •g ,
where Perm(X) denotes the group of all permutations (i.e., bijections) of the set X. If n =
{1, 2, . . . , n}, then Sn = Perm(n) is the symmetric group on nobjects, and has order n!.
Given such an action of G on X, we have the following de¬nitions:
Stab• (x) = {g ∈ G : •g (x) = x},
Orb• (x) = {y ∈ X : for some g ∈ G, y = •g (x)},
X G = {x ∈ X : gx = x ∀g ∈ G}.
Then Stab• (x) is called the stabilizer of x and is often denoted StabG (x) when the action is
clear, while Orb• (x) is called the orbit of x and is often denoted OrbG (x). X G is called the
¬xed point set of the action.
Theorem 2.12. Let • be an action of G on X, and x ∈ X.
(1) Stab• (x) is a subgroup of G. Hence if G is ¬nite, then so is Stab• (x) and moreover
by Lagrange™s Theorem,  Stab• (x)  G.
(2) There is a bijection
G/ Stab• (x) ←’ Orb• (x); g Stab• (x) ” g · x = •g (x).
Furthermore, this bijection is Gequivariant in the sense that
hg Stab• (x) ” h · (g · x).
In particular, if G is ¬nite, then so is Orb• (x) and we have
 Orb• (x) = G/ Stab• (x).
The distinct orbits partition X into a disjoint union of subsets,
X= Orb• (x).
distinct
orbits
Equivalently, there is an equivalence relation ∼ on X for which the distinct orbits are the
G
equivalence classes and given by
x∼y ⇐’ for some g ∈ G, y = g · x.
G
Hence, if X is ¬nite, then
X =  Orb• (x)
distinct
orbits
16 2. RECOLLECTIONS AND REFORMULATIONS ON BASIC GROUP THEORY
This theorem is the basis of many arguments in Combinatorics and Number Theory as well
as Group Theory. Here is an important example, often called Cauchy™s Lemma.
Theorem 2.13 (Cauchy™s Lemma). Let G be a ¬nite group and let p be a prime for which
p  G. Then there is an element g ∈ G of order p.
Proof. Let
X = Gp = {(g1 , g2 , . . . , gp ) : gj ∈ G, g1 g2 · · · gp = eG }.
Let H be the group of all cyclic permutations of the set {1, 2, . . . , p}; this is a cyclic group of
order p. Consider the following action of H on X:
γ · (g1 , g2 , . . . , gp ) = (gγ ’1 (1) , gγ ’1 (2) , . . . , gγ ’1 (p) ).
It is easily seen that this is an action. By Theorem 2.12, the size of each orbit must divide
H = p, hence must be 1 or p since p is prime. On the other hand,
X = Gp ≡ 0 (mod p),
since p  G. Again by Theorem 2.12, we have
X =  OrbH (x),
distinct
orbits
and hence
 OrbH (x) ≡ 0 (mod p).
distinct
orbits
But there is at least one orbit of size 1, namely that containing e = (eG , . . . , eG ), hence,
 OrbH (x) ≡ ’1 (mod p).
distinct
orbits not
containing e