Abstract Algebra

Show that a group G is abelian if for all elements a and b in G, a * b = b * a.

Demonstrate that the group of symmetries of a square in 3D is not abelian by finding elements a and b such that a * b ≠ b * a.

Let's examine the difference between a left coset and a right coset using the group $S_4$ and a subgroup $H$. Compute the left cosets and right cosets of a specific element and show that they are generally different in a non-abelian group.

In an abelian group, demonstrate that left and right cosets must be the same by using the group Z8 and the subgroup {0, 4}. Compute left cosets and right cosets for a specific element and confirm their equality.

Multiply the two permutations given in cycle notation: A with three disjoint 3-cycles, and B as one 9-cycle.

Let $G$ be the group of matrices $\begin{bmatrix} a & b \\ 0 & c \end{bmatrix}$ where $a$ and $c$ are non-zero real numbers, and $b$ is any real number. Define a homomorphism $F: G \to \mathbb{R}^*$ by $F(\begin{bmatrix} a & b \\ 0 & c \end{bmatrix}) = ac$. Prove that $F$ is a homomorphism and determine its kernel $K$. Using the First Isomorphism Theorem, show that $G/K \cong \mathbb{R}^*$.

Suppose $\phi: R \to S$ is a ring homomorphism. Prove that there exists a unique ring isomorphism $\psi: R/\text{ker}(\phi) \to \text{Im}(\phi)$ such that $\psi(r + \text{ker}(\phi)) = \phi(r)$.

Determine whether the following mappings represent a function: Inputs: \{2, 4, 5\}, Outputs: \{-1, 0, 4\}.

Determine whether the following mappings represent a function: Inputs: {-4, 0, 8}, Outputs: {1, 2, 5, 7}.

Determine whether the following mappings represent a function: Inputs: {-3, -2, 0, 4}, Outputs: {-7, -5, -3}.

Describe a bijection $heta$ from the set mathbb{Z} of integers to its proper subset $E$, such that $E$ is the set of multiples of two of each integer. Show that the function $heta(x) = 2x$ is bijective.

Describe an injection from a set $X = \{x_1, x_2, \ldots, x_n\}$ with $n$ elements to the set $\mathbb{Z}$ of integers, where $\theta(x_i) = i$.

Let $\theta: \mathbb{R} \to \mathbb{N}_0$ be a mapping given by $\theta(x) = |x|$. Is $\theta$ injective? Explain.

List the elements of the complete inverse image $\theta^{-1}(5)$ under the mapping $\theta(x) = |x|$.

Let $g: \mathbb{Z} \to \mathbb{Z}_5$ such that for any integers $m$ and $n$, $g(mn) = g(m)g(n)$. Does this hold under modular arithmetic?

Find the inverse of 2 modulo 9.

Determine if 3 has an inverse modulo 9.

Find the inverse of 4 modulo 9.

Determine if 6 has an inverse modulo 9.

Find the inverse of 8 modulo 9.