S2, C3: Groups and Subgroups

Question 1

Definition 3.1: Binary Operation

Answer 1

Let A be a non-empty set. A binary operation * on A is a rule which, for each ordered pair (a, b) of elements of A, determines a unique element a * b of A.
Equivalently, a binary operation is a function
*: A x A → A.
When * is a binary operation on the set A we often say that A is closed under ; that is, ab ∈A for all a, b ∈ A.
Examples: +, -, x, etc.

Question 2

Definition 3.3: A Group

Answer 2

G is a group under * if:
G1: Closure. That is, ab ∈A for all a, b ∈ A.
G2: Associativity. i.e. a(bc) = (ab)*c for all a,b,c∈G.
G3: Neutral Element. (unique)
G4: Inverses. (unique)

Question 3

Definition 3.4: Abelian Group

Answer 3

A group G is said to be Abelian if ab=ba for every a, b∈G^2. (Commutativity)

Question 4

Proposition 3.9: Under what conditions is Zp{0}, where p is prime, a group?

Answer 4

If p is prime then Zp{0} is an abelian group under

multiplication modulo p.

Question 5

Definition 3.13: General Linear Group

Answer 5

For any field F, the invertible 2x2 matrices over F form the General Linear Group GL2(F)

Question 6

Definition 3.14. Order of a group

Answer 6

The order of a group, written |G|, is the number of elements in G. This is either infinite or a positive integer.

Question 7

Latin Square Property

Answer 7

If G is a finite group, every element g∈G appears once in every row and column in the Cayley table for G.

Question 8

Definition 3.7: Subgroups

Answer 8

Let G be a group. A subset H of G is a subgroup of G if H is a group using the same binary operation as G.
SG1: H (is not equal to) ∅.
SG2: gh∈H for all g, h ∈ H. (H is closed under binary operation)
SG3: h^(−1) ∈ H for all h ∈ H.

Question 9

Definition 3.27: Groups of Symmetries.

Answer 9

Let A be a geometrical figure in R^2 with centre at the
origin and let f be a rotation or reflection in the orthogonal group O2. The set f(A) = {f(a) : a ∈ A} is called the image of A. If f(A) = A then f is said to be a symmetry of A.

Question 10

Definition 3.33. Cartesian Products of Groups.

Answer 10

Let G and H be groups. The cartesian product of GxH consists of all the ordered pairs (g, h), where g∈G and h∈H.
We define a binary operation on GxH componentwise.

Question 11

Definitions 3.35. Homomorphism, Isomorphism.

Answer 11

Let G and H be groups and f: G→H be a function. Then f is said to be a homomorphism if f(xy)=f(x)f(y) for all x, y ∈ G.
A homomorphism which is bijective is said to be an isomorphism.
We say that two groups G and H are isomorphic if there is an isomorphism f:G→H.