Chapter 1

Question 1

Definition 1.1. A group is a set G together with a binary operation (de- noted as multiplication) such that the following axioms hold:

Answer 1

Closure: G is closed for the operation: x, y ∈ G =⇒ xy ∈ G;

Associativity: (xy)z = x(yz) for all x, y, z ∈ G;

Identity: there exists an element e ∈ G (called the identity of G) such that xe = ex = x for all x ∈ G;

Inverses: for every element x ∈ G there exists an element x−1 ∈ G (called the inverse of x) such that xx−1 = x−1x = e.

Question 2

Theorem 1.2. The following statements are true for any group G.
(i) (a^m)(a^n)=
(ii)(a^m)^n=
…(There are 7 in total)

Answer 2

a^(n+m), for all a∈G and all m,n∈Z
a^(mn), for al la∈G and all m,n∈Z
(iii) The identity element e is unique.
(iv) The inverse of any element is unique.
(v) (a^(−1))^(−1) =a for all a∈G.
(vi) (ab)^(-1) = b^(-1)a^(−1) for all a, b ∈ G.
(vii) The cancellativity laws hold:
for all a,x,y ∈ G.
ax = ay ⇒ x = y xa = ya ⇒ x = y

Question 3

The order of a group G (denoted by |G|) is…

Answer 3

The number of elements of G.

Question 4

Definition 1.4. Let G be a group. Two elements x, y ∈ G are said to commute if…
If all pairs of elements x,y ∈ G commute, G is said to be

Answer 4

xy=yx
Abelian

Question 5

Let G be a group. Two elements x, y ∈ G are said to be conjugate if

Answer 5

y = a^(-1)xa, for some a ∈ G; this will sometimes be denoted x ∼ y.

Question 6

Definition 1.6. Two groups G and H are said to be isomorphic (G ∼= H) if…

What kind of relation is this???

Answer 6

There is a bijection f : G −→ H, which agrees with their structures, in the sense that
f(xy) = f(x)f(y),
for all x, y ∈ G. Any such mapping is called an isomorphism.
An equivalence relation.

Question 7

The sets Z of integers, Q of rationals, R of reals, and C of complex numbers all form…

Answer 7

abelian groups under addition.

Also the set
Zn ={0,1,…,n−1}
under addition modulo n forms a group.

Question 8

if 0 is removed from the sets Q, R, C, then

Answer 8

They form multiplicative data.

Question 9

The set Zn{0} is a multiplicative group if and only if

Answer 9

n is a prime

Question 10

A permutation on X is

Answer 10

a bijection from X onto X

Question 11

The set of all permutations of X is denoted

Answer 11

S_X

Question 12

SX is a group; it is usually called the symmetric group on X. Its operation is…

Answer 12

Composition of mappings.

Question 13

Assume that X = {1, . . . , n}. Then the order of SX is…

Answer 13

n!

Question 14

Two cycles are disjoint if…

Answer 14

no element is moved by both of them; disjoint cycles commute.
It is well known that every permutation from Sn can be written as a product of disjoint cycles, and that this decom- position is unique up to the order of factors

Question 15

A cycle of length 2 is called

Answer 15

a transposition.
Note that every cycle can be
written as a product of transpositions
However this product is NOT unique

Question 16

Conjugation works particularly nicely in symmetric groups, and this is worth recording as it will be used in many concrete examples throughout.
The basic observation for this is the following formula for conjugating a cycle γ=(i1 i2 … ir)∈Sn by a permutation π∈Sn: π^(-1)γπ=

Answer 16

(i1π i2π . . . irπ)

Question 17

We say that two permutations τ and σ have the same disjoint cycle struc- ture if

Answer 17

The decompositions of σ and τ contain the same numbers of cycles of each length.
Thus, for example, the permutations (1 2 3)(4 5) and (1 2 5)(3 4) have the same disjoint cycle structure, while (1 2 3)(4 5) and (1 2)(3 4) do not.

Question 18

Theorem 2.1. Two permutations σ, τ ∈ Sn are conjugate in Sn if and only if

Answer 18

they have the same disjoint cycle structure.
Try proving this! pg 11+12

Question 19

What is a General linear group GL(n, F)

Answer 19

The set of non matrices with entries for the field F with the operation matriculates multiplication. It is non-abelian.

Question 20

Describe the quaternion group:

Answer 20

This group consists of elements 1, −1, i, −i, j, −j, k, −k.
Here the first four of them are the usual complex numbers (i being the imaginary unity). j and k are two further imaginary units. They multiply according to the rules:

i^2 =j^2 =k^2 =−1 , ij = −ji = k , jk=−kj=i, ki = −ik =j.