Chapter 3 Flashcards
Def 3.1.1: What is a group?
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
Def 3.1.10: What is the order of an element?
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
Def 3.1.5: What is the order of a group?
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) = ∞
Def 3.3.1: What is the set S_A?
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.
Def 3.3.3: What is S_n?
For any positive integer n, S_n is the set of all permutations of {1, 2, … , n} with the inary operation of composition.
Def 3.3.6: What is a cycle?
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.
Def 3.3.12: What is a transposition?
A transposition is a cycle of length two, σ = (a_1 a_2), a_1 != a_2.
Def 3.3.15: What is an even/odd permutation?
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.
Def 3.5.3: What is D_n?
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}
Def 3.6.1: What is ?
Let G be a group and g be an element of G. We define
Def 3.6.2: What is a generator of G?
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}
Def 3.7.1: What is a subgroup?
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.
Def 3.7.9: What does represent?
Let G be a group and g be any element of G. The set is called the subgroup generated by g.
Def 3.7.12: What is A_n?
The alternating group on n symbols, is the subgroup of S_n consisting of all even permutations.
Def 3.8.1: What is GL_n(R)?
The set of all n x n matrices with real entries, which are invertible.
Def 3.8.4: What is GL_n(Z)?
For any positive integer n, it represents the set of all n x n matrices with integer entries and a determinant of ±1.
Def 3.9.1: What is a left coset of G? What does G/H represent?
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)
Def 3.9.9: How do you calculate the number of distinct left cosets?
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
Def 3.10.1: What is the orbit of x under G?
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?
Def 3.10.4: What is the stabilizer of x?
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?
Def 3.11.3: What is the set A^g?
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?
Def 3.12.1: What is an isomorphism? How is it denoted?
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.
What is O_n(R)?
{A ∈ Mn(R) | (A^T)A = I_n = A(A^T)
What is the ax+b group?
The set
{ [a b]
[0 1 ]
| a, b ∈ R, a != 0}
how to tell if 2 left cosets are equal
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.
What is Lagranges Theorem?
o(G) = o(H) [G : H]
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).
idk
What is the Orbit Stabilizer Theorem?
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)
What is Burnsides Lemma?
A/G = 1/(#G) *(the sum over all g ∈ G of A^g)
What is a ring?
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 +.
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.
just rmbr that
Let R be a ring. What does R need to be a field?
1) multiplicative identity 1 != 0
2) commutative
3) every element other than 0 is a unit
What is a zero divisor?
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.
What is an integral domain?
A ring R is an integral domain if it is commutative, has an identity, and has no zero divisors.
