Mathematics II · Unit 9 · 5 hrs
Groups and Subgroups
Exam-focused notes for Groups and Subgroups (Mathematics II, MTH168): what the TU syllabus asks and how it has actually been tested, with 6 solved past questions from this unit.
What this unit covers
- Binary Operations
- Groups
- Subgroups
- Cyclic Groups
Groups
Define group. Show that the set of integers is a group with respect to addition operation. [5]
Definition: A non-empty set G together with a binary operation \ is called a group, denoted (G, \), if the following axioms are satisfied: 1. Closure: For all a, b ∈ G, a \ b ∈ G 2. Associativity: For all a, b, c ∈ G, (a \ b) \ c = a \ (b \ c) 3. Identity: ...
Full solved answer →Define group. Show that the set of integers is not a group with respect to subtraction operation. [5]
A group is a non-empty set G together with a binary operation \ such that the following four axioms (properties) are satisfied: 1. Closure: For all a, b ∈ G, a \ b ∈ G. 2. Associativity: For all a, b, c ∈ G, (a \ b) \ c = a \ (b \ c). 3. Identity Element: T...
Full solved answer →Let an operation $*$ be defined on $\mathbb{Q}^+$ by $a * b = \frac{ab}{2}$. Then show that $\mathbb{Q}^+$ forms a group. [5]
A non-empty set G with a binary operation ∗ is called a group if it satisfies the following four axioms: 1. Closure 2. Associativity 3. Existence of Identity 4. Existence of Inverse Here, the operation is defined on Q⁺ (the set of all positive rationals) as...
Full solved answer →Define group. Show that the set of all integers, Z forms group under addition operation. [5]
A non-empty set G together with a binary operation \ is called a group if the following axioms are satisfied: Axiom Condition ------------------ G1: Closure For all a, b ∈ G, a \ b ∈ G G2: Associativity For all a, b, c ∈ G, (a \ b) \ c = a \ (b \ c) G3: Ide...
Full solved answer →Define group. Show that (Z, .) doesn't form a group. [5]
A group is an algebraic structure consisting of a non-empty set G together with a binary operation such that the following four axioms are satisfied: Axiom Condition Description ------------------------------- G1 Closure For all a, b in G, a b is in G G2 As...
Full solved answer →Binary Operations
Define binary operation. Determine whether the binary operation * is associative or commutative or both where * is defined on ℚ by letting $x * y = \frac{x + y}{3}$ [5]
A binary operation on a non-empty set S is a rule that assigns to each ordered pair (a, b) of elements of S a unique element a\b in S. Formally, a mapping \ : S x S - S is called a binary operation on S if: - For all a, b in S, a\b is in S (closure property...
Full solved answer →Make Unit 9 stick
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