T1: Lectures 1-5

Question 1

Definition of a group

Answer 1

A triple including a set, binary operation and identity element.

Question 2

Give two additional criteria for a group

Answer 2

The binary operation is associative and for each element there exists an inverse.

Question 3

Define the symmetric group S_n

Answer 3

Characterises the permutations of n elements.

Order n!

Question 4

Give the presentation of S_3

Answer 4

Where s=(1 2), t=(2 3) we have:

<s,t|s^2=t^2=e, sts=tst>

Question 5

Define the dihedral group D_n

Answer 5

The set of symmetries of a regular n-gon inscribed on the unit circle with a point at (1,0)

Size 2n

Question 6

How many rotation and reflections are in D_n (and hence give the size/order of the group)

Answer 6

n rotations and n reflections (hence, 2n).

Question 7

Define the quaternion group Q_8

Answer 7

The set of elements +_{1,i,j,k} with i^2=j^2=k^2=ijk=-1

Question 8

Define GL_n(V)

Answer 8

The set of n x n matrices with non zero determinant that characterise a linear map from V to V.

Question 9

The function φ mapping G to H two groups (G, ⋅) and (H, *) is a homomorphism if..?

Answer 9

φ(g_1)*φ(g_2)
= φ(g_1 ⋅ g_2)

Question 10

What are the two key ways to verify a homomorphism?

Answer 10

Check the homomorphism relation, or verify that the map preserves the presentation of the original group.

Question 11

Define a representation of a group (π,V)

Answer 11

A pair (π,V) where v is a vector space, and the map π: G to GL_n(V) is a group homomorphism
π(g)π(h)=π(gh) for all g,h in G.

Question 12

What is the permutation representation of the symmetric group S_n?

(column vector)

Answer 12

(π, C^n) where we have an n dimensional vector with entries holding:

π(σ)(z_n)=(z_(σ^(-1)(n))

Question 13

What is the permutation representation of the symmetric group S_n?

(unit vector)

Answer 13

(π, C^n) where we have an n term sum with entries holding:

π(σ)(x_1e_1+ … +x_ne_n)
= x_1e_σ(1)+ … + x_ne_σ(n).

Question 14

Define the trivial representation of a group G

Answer 14

(π,V) where π: g to Id
for all g in G

Question 15

What is the dimension of a representation?

Answer 15

The dimension of the underlying vector space.

Question 16

What is the defining rep for D_n

Answer 16

The pair of matrices ρ(r) and ρ(s)

-sin top right

-1 top left

Question 17

How can we show that a set of matrices represent a group

Answer 17

The same way as showing a homomorphism: check the homomorphism condition or verify the presentation.

Question 18

How does a subspace form a subrepresentation?

Answer 18

Let (π,V) be a rep of G. A subspace W⊆V forms a subrepresentation (π,W) if for all π(g)w the result is inside W.

Question 19

Define an irreducible representation

Answer 19

A rep (π,V) is irreducible if it has no non-trivial subrepresentations: (π,0) and (π,V).

Question 20

When finding subgroups, what is the condition on the order of elements and subgroups?

Answer 20

The order of any element and any subgroup must divide the order of the group.

Question 21

How can we determine a conjugacy class?

Answer 21

conjugate all elements of the group between the generators and match up corresponding (or cyclic) results.

Question 22

What is a condition on a conjugacy class?

Answer 22

All elements have the same order.

Question 23

What is the order of a group?

Answer 23

The number of elements

Question 24

What is the order of an element in a group?

Answer 24

The power you must raise it to to return the identity.

Question 25

Consider the permutation rep of S_n and give the two subspaces for the subreps

Answer 25

W = complex number * n-dim column vector of ones

W^⊥ = n dim column vector st sum of elements = 0

Question 26

How do the subspaces in the sub reps of S_3 relate to one another and the whole vector space of S_3?

Answer 26

W and W^⊥ are orthogonal and their direct sum = V

Question 27

How can we prove that a representation is irreducible.

Answer 27

Consider an irreducible subrep of the rep in question and show that the vector spaces must be equal.

Question 28

Complete the remark: “Every representation of a finite group spanning a complex vector space…

Answer 28

…can be decomposed into a direct sum of irreducble reps.

Question 29

Consider two reps of a group G: (π_1,V) and (π_2,W). The linear map T:V to W is a hom if:

Answer 29

Tπ_1(g) = π_2(g)T for all g in G

(alt T(π_1(g)v) = π_2(g)T(v) )

Question 30

Hom_G (V,W) denotes..

Answer 30

The vector space of all G homomorphisms from V to W.

Question 31

Give the condition for a G-hom to be an isomorphism

Answer 31

The linear map T (and hence T^-1) must be invertible.

Question 32

Given T is a G-hom between (π_1,V) and (π_2,W), define a subrep for each of the latter.

Answer 32

(π_1, ker(T)) and (π_2, Im(T)) are subreps of (π_1,V) and (π_2,W) respectively.

Question 33

Define Ker(T)

Answer 33

The set of vectors for which Tv = 0

Question 34

Define Im(T)

Answer 34

The set of vectors w for which w = Tv

Question 35

State Schur’s Lemma (part ii)

Answer 35

If (π,V) is a finite-dimensional irreducible representation of group G, then Hom_G (V)=CId

Question 36

Define Abelian Group

Answer 36

A group for which the group operation is commutative.

Question 37

State the first isomorphism theorem

Answer 37

For T: V to U

The map taking quotient space V/Ker(T) to U is isomorphic to taking V/Ker(T) to Im(T)

Question 38

Consider first iso theorem. How do the dim kernel and image relate to the dim of V

Answer 38

dim ker + dim im = dimV

Question 39

What are the kernel and image of a Hom f: G to G’ between groups

Answer 39

ker(f) = g in G, f(g) = e

im(f) = g in g, f(g)

Question 40

State Schur’s Lemma (part i)

Answer 40

For T in HomG(V,W) between two irreducible finite complex reps, either T is an iso, or T = 0

Question 41

State Schur’s Lemma (part iii)

Answer 41

dimHomG(V,W) for a irr rep is 1 if V and W are isomorphic or 0 else.

Question 42

What is special about Abelian groups?

Answer 42

Every finite dim irr rep is one-dimensional.