Chapter 3

Question 1

Def 3.1.1: What is a group?

Answer 1

A group is a non-empty set G, together with a binary operation satisfying the following conditions:

1) The binary operation is associative
2) The set G has an identity
3) Every element has an inverse, also in G

Question 2

Def 3.1.10: What is the order of an element?

Answer 2

If g^k != e for every positive integer k, we say that g has infinite order, o(g) = ∞

If g^k = e for some k, we define the order of g to be the least such k, o(g) = k

Question 3

Def 3.1.5: What is the order of a group?

Answer 3

If group G is finite, it has finite order, and o(G) = the number of elements in G.

If G is infinite, we write o(G) = ∞

Question 4

Def 3.3.1: What is the set S_A?

Answer 4

Let A be a non-empty set. S_A is the set of all functions f: A → A which are bijective.

For each non-empty set A, S_A with composition of functions is a group.

Question 5

Def 3.3.3: What is S_n?

Answer 5

For any positive integer n, S_n is the set of all permutations of {1, 2, … , n} with the inary operation of composition.

Question 6

Def 3.3.6: What is a cycle?

Answer 6

Let a_1, …, a_k, be distinct elements of 1, 2, .., n. We let (a_1 … a_k) denote the element of S_n which maps a_1 to a_2, a_2 to a_3, …, a_k to a_1 and maps every other element of {1, 2, …, n} to itself. We refer to such a permutation as a cycle and we say it has length k.

Question 7

Def 3.3.12: What is a transposition?

Answer 7

A transposition is a cycle of length two, σ = (a_1 a_2), a_1 != a_2.

Question 8

Def 3.3.15: What is an even/odd permutation?

Answer 8

Any permutation which can be expressed as a product of an even number of transpositions is an even permutation, similar for odd.

We define sgn(π) to be 1 is π is even, and -1 if it is odd.

Question 9

Def 3.5.3: What is D_n?

Answer 9

The symmetries of a regular n-gon.

D_n = {e, r, r^2, …, r^(n-1), j, rj, r^(2)j, …, r^(n-1)j}

Question 10

Def 3.6.1: What is ?

Answer 10

Let G be a group and g be an element of G. We define

Question 11

Def 3.6.2: What is a generator of G?

Answer 11

We say that G is cyclic, and that g is a generator of G is there is an element g in G such that
G = g = {g^n | n ∈ Z}

Question 12

Def 3.7.1: What is a subgroup?

Answer 12

Let G be a group. A subgroup of G is a non-empty subset H of G which, using the same binary operation as for G, becomes a group itself.

Question 13

Def 3.7.9: What does represent?

Answer 13

Let G be a group and g be any element of G. The set is called the subgroup generated by g.

Question 14

Def 3.7.12: What is A_n?

Answer 14

The alternating group on n symbols, is the subgroup of S_n consisting of all even permutations.

Question 15

Def 3.8.1: What is GL_n(R)?

Answer 15

The set of all n x n matrices with real entries, which are invertible.

Question 16

Def 3.8.4: What is GL_n(Z)?

Answer 16

For any positive integer n, it represents the set of all n x n matrices with integer entries and a determinant of ±1.

Question 17

Def 3.9.1: What is a left coset of G? What does G/H represent?

Answer 17

Let G be a group, H a subgroup, and g an element of G. The left coset of H in G determined by g is gH = { gh | h ∈ H}.
G/H denotes the collection of left cosets. (its elements are subsets of G)

Question 18

Def 3.9.9: How do you calculate the number of distinct left cosets?

Answer 18

Let G be a group and H a subgroup. The index of H in G is the number of distinct left cosets and is denoted by [G : H] = #G/H

Question 19

Def 3.10.1: What is the orbit of x under G?

Answer 19

Let A be a non-empty set, and let G be a subgroup of S_A. For any x in A, the orbit of x under G is
G(x) = {g(x) | g ∈ G}

AKA: what are the other points of A you can move X to, using elements of G?

Question 20

Def 3.10.4: What is the stabilizer of x?

Answer 20

Let A be a non-empty set, and let G be a subgroup of S_A. For any x in A, we define the set
G_x = {g ∈ G | g(x) = x}

AKA: what elements of G leave x where it is?

Question 21

Def 3.11.3: What is the set A^g?

Answer 21

Let A be a set and G be a subgroup of S_A. For each element g in G, let A^g be the points of A which are fixed by g; that is,
A^g = {x ∈ A | g(x) = x}

AKA: Basically the opposite of G_x, given an element g, what x’s stay where they are?

Question 22

Def 3.12.1: What is an isomorphism? How is it denoted?

Answer 22

Let G and H be groups. A function n f : G → H is called an isomorphism if
1) f is a bijection
2) for all a, b in G, f(ab) = f(a) f(b)
If such a map exists, we say that G and H are isomorphic and write G ∼= H.

Question 23

What is O_n(R)?

Answer 23

{A ∈ Mn(R) | (A^T)A = I_n = A(A^T)

Question 24

What is the ax+b group?

Answer 24

The set
{ [a b]
[0 1 ]
| a, b ∈ R, a != 0}

Question 25

how to tell if 2 left cosets are equal

Answer 25

Let G be a group and let H be a subgroup. For g1, g2 in G,
define g1 ∼ g2 if g1^(-1)g2 is in H. Then ∼ is an equivalence relation on G and,
for any g in G, the equivalence class of g is exactly the left coset gH.
That
is, for any g1, g2 in G, g1H = g2H if and only if g1^(-1)g2 is in H.

Question 26

What is Lagranges Theorem?

Answer 26

o(G) = o(H) [G : H]

Question 27

If g is any element of G, the left coset of Gx which contains g, namely
gGx, is exactly the set of elements h in the group G such that h(x) =
g(x).

Answer 27

idk

Question 28

What is the Orbit Stabilizer Theorem?

Answer 28

Let A be a non-empty set, and let G be a finite subgroup of S_A.  Then, for any x in A, we have
#G(x) · o(G_x) = o(G)

Question 29

What is Burnsides Lemma?

Answer 29

A/G = 1/(#G) *(the sum over all g ∈ G of A^g)

Question 30

What is a ring?

Answer 30

A ring is a non-empty set R with two binary operations, denoted + and ·, satisfying the following conditions.

1) (R, +) is an abelian group
2) · is associative.
3) The operation · is distributive over +.

Question 31

In a ring (R, +, ·) with identity, an element a in R has
an inverse if it has an inverse with respect to the operation ·. In this case,
we write its inverse as a^(-1). We also say that a is a unit of R if it has an
inverse.

Answer 31

just rmbr that

Question 32

Let R be a ring. What does R need to be a field?

Answer 32

1) multiplicative identity 1 != 0
2) commutative
3) every element other than 0 is a unit

Question 33

What is a zero divisor?

Answer 33

Let R be a commutative ring with identity. A non-zero element a in R is a zero divisor if there is another non-zero element b in R
such that ab = 0.

Question 34

What is an integral domain?

Answer 34

A ring R is an integral domain if it is commutative, has an identity, and has no zero divisors.