🌀 abstract algebra

1. **Problem Statement:** Prove by the principle of mathematical induction that $$3^{2n+1} + (-1)^n 2 \equiv 0 \pmod{5}$$ for all positive integers $n$. 2. **Principle of Mathemati

1. **Define Group, Semigroup, Monoid, Subgroup, Abelian Group, Cyclic Group, and Lattice with Properties** - **Group:** A set $G$ with a binary operation $\cdot$ is a group if it s

1. **Problem Statement:** We need to show that the relation $R = \{(a,b) \mid a \equiv b \pmod{m}\}$ is an equivalence relation on $\mathbb{Z}$ and verify that if $x_1 \equiv y_1$

1. **Problem statement:** Show that for a group $G$ and fixed element $a \in G$, the set $$H_a = \{x \in G : xa = ax\}$$

1. The problem is to understand the syllabus and topics covered in the DSC-14 Ring Theory course. 2. The course covers foundational and advanced topics in ring theory, including de

1. **Problem Statement:** Draw addition (\oplus) and multiplication (\otimes) tables for the set $\{2,4,6,8\}$ modulo 9.

1. لنفترض أن السؤال هو تحويل مجموعة معينة إلى صورة باستخدام مفهوم الـ cosets. 2. الـ coset هو مجموعة من العناصر التي تُكوّن عندما نضيف عنصرًا معينًا إلى كل عنصر في مجموعة فرعية.

1. **State the problem:** We have a map $\rho: \mathbb{Z}[i] \to \mathbb{Z}/10\mathbb{Z}$ defined by $\rho(a + bi) = [a + 7b]$. We want to find the kernel of this homomorphism. 2.

1. **Problem Statement:** Prove that all subgroups of a cyclic group are fully invariant subgroups of that cyclic group. 2. **Definitions:**

1. **Problem Statement:** We have a map $f : (G, \Delta) \to (\mathbb{R} \setminus \{0\}, \times)$ defined by $f(x) = x - \alpha$.

1. **Problem Statement:** Consider the group $\mathbb{Z}_5 = \{0,1,2,3,4\}$ under addition modulo 5. Find the order of $\mathbb{Z}_5$ and determine whether each element is a genera

1. **Problem statement:** We have a group $G$ and a subgroup $G'$ generated by all elements of the form $aba^{-1}b^{-1}$ for $a,b \in G$. We want to prove:

1. **State the problem:** We want to evaluate the expression $$\alpha = \left(2 \cdot \left[ 2 \cdot (3^{-1}) - 2^{-1} \right] - \left[ 2 \cdot (-2)^{-1} + 2^{-1} \right]^{-1}\righ

1. Abstract algebra is a branch of mathematics that studies algebraic structures such as groups, rings, and fields. 2. The fundamental concept is a \textbf{group}, which is a set e

1. **Problem Statement:** Prove that every group $G$ is isomorphic to a group of permutations on $G$ itself, and that if $\varphi: G \to G'$ is an isomorphism, then its inverse $\v

1. The problem asks to explain what a bilinear function is when defined from a Cartesian product to an abelian group $G$. 2. A bilinear function $f$ is a function defined on the Ca

1. **Problem Statement:** Show that $$\mathbb{Q} = 1 - (\mathbb{Q} - \mathbb{Z})$$ in the context of $$\mathrm{Hom}_{\mathbb{Z}}\mathbb{Z}$$. 2. **Understanding the notation:** Her

1. Let's clarify the problem statement to understand what "S4" refers to. 2. If "S4" refers to a mathematical concept, such as the symmetric group on 4 elements, please specify the

1. **Problem (a): Determine the order of the group $G=\mathbb{Z}_6$ and the order of a subgroup in $G$.** The group $\mathbb{Z}_6$ consists of integers modulo 6 under addition: $\{

1. **Problem statement:** Find all ring homomorphisms from $\mathbb{Z}_6$ to $\mathbb{Z}_6$ and from $\mathbb{Z}_{20}$ to $\mathbb{Z}_{30}$. 2. **Recall:** A ring homomorphism $\va

1. The problem is to understand the definition and properties of a finite field, focusing on the addition operation. 2. A field $F$ is a non-empty set with two operations: addition