FIRST COURSE IN ABSTRACT ALGEBRA
ACTUAL TEST PAPER QUESTIONS AND
SOLUTIONS GRADED A+
◉ Well-Defined.
Answer: A function (f: A → B) where for each a in the domain there is
a unique b in the codomain such that f(a) = b. In other words, if a = c,
then f(a) = f(c).
◉ Image (or Range).
Answer: The subset of the codomain that a function maps to.
◉ Injective (One-to-One).
Answer: If f(a1) = f(a2), then a1 = a2.
◉ Surjective (or Onto).
Answer: For all b in the codomain, there exists some a in the domain
such that f(a) = b.
◉ Bijective.
Answer: When a function is both injective and surjective.
,◉ U(n).
Answer: {a ∈ Zn | gcd(a, n) = 1}
◉ Three Properties of a Group.
Answer: 1. The binary operation is associative.
2. There exists an identity element.
3. Each element has an inverse.
◉ Order of an Element.
Answer: The smallest positive integer such that x^n = e.
◉ Order of a Group.
Answer: The number of elements in a group.
◉ Generator.
Answer: A set of elements from a group G whose products can form
the entire group G.
◉ Cyclic Group.
Answer: A group that can be generated from a single element.
◉ General Linear Group (GL_n(F)).
, Answer: {A ∈ Mat(n,F) | det(A) != 0}
The set of n x n matricies with entries in the field F and a nonzero
determinant.
◉ Special Linear Group (SL_n(F)).
Answer: {A ∈ GL_n(F) | det(A) = 1}
The set of n x n matrices with entries in the field F and determinant
equal to 1.
◉ Orthogonal Group (O_n(F)).
Answer: {A ∈ GL_n(F) | A^T A = A A^T = I}
The set of n x n orthogonal matricies with entries in field F and
nonzero determinant.
◉ The Special Orthogonal Group (SO_n(F)).
Answer: {A ∈ SL_n(F) | A^T A = A A^T = I}
The set of n x n matrices with entries in field F and determinant
equal to 1.
◉ Homomorphism.
Answer: A map Φ: G → H where Φ(x * y) = Φ(x) * Φ(y).
◉ Kernel of a Homomorphism (ker Φ).
ACTUAL TEST PAPER QUESTIONS AND
SOLUTIONS GRADED A+
◉ Well-Defined.
Answer: A function (f: A → B) where for each a in the domain there is
a unique b in the codomain such that f(a) = b. In other words, if a = c,
then f(a) = f(c).
◉ Image (or Range).
Answer: The subset of the codomain that a function maps to.
◉ Injective (One-to-One).
Answer: If f(a1) = f(a2), then a1 = a2.
◉ Surjective (or Onto).
Answer: For all b in the codomain, there exists some a in the domain
such that f(a) = b.
◉ Bijective.
Answer: When a function is both injective and surjective.
,◉ U(n).
Answer: {a ∈ Zn | gcd(a, n) = 1}
◉ Three Properties of a Group.
Answer: 1. The binary operation is associative.
2. There exists an identity element.
3. Each element has an inverse.
◉ Order of an Element.
Answer: The smallest positive integer such that x^n = e.
◉ Order of a Group.
Answer: The number of elements in a group.
◉ Generator.
Answer: A set of elements from a group G whose products can form
the entire group G.
◉ Cyclic Group.
Answer: A group that can be generated from a single element.
◉ General Linear Group (GL_n(F)).
, Answer: {A ∈ Mat(n,F) | det(A) != 0}
The set of n x n matricies with entries in the field F and a nonzero
determinant.
◉ Special Linear Group (SL_n(F)).
Answer: {A ∈ GL_n(F) | det(A) = 1}
The set of n x n matrices with entries in the field F and determinant
equal to 1.
◉ Orthogonal Group (O_n(F)).
Answer: {A ∈ GL_n(F) | A^T A = A A^T = I}
The set of n x n orthogonal matricies with entries in field F and
nonzero determinant.
◉ The Special Orthogonal Group (SO_n(F)).
Answer: {A ∈ SL_n(F) | A^T A = A A^T = I}
The set of n x n matrices with entries in field F and determinant
equal to 1.
◉ Homomorphism.
Answer: A map Φ: G → H where Φ(x * y) = Φ(x) * Φ(y).
◉ Kernel of a Homomorphism (ker Φ).
