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MathidiaMathematics Academy
Groups and symmetry
Lesson 16 of 75
Introduction

Groups and examples

Objective. Verify group axioms in varied structures.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Groups and examples formalise symmetry through a set G with a binary operation satisfying closure, associativity, an identity element e with ge = eg = g, and inverses g⁻¹ with gg⁻¹ = e for every g ∈ G. Common examples include (ℤ, +) with identity 0, and invertible n×n matrices under multiplication, GL(n,ℝ), with identity the identity matrix. Checking the axioms matters: (ℕ, +) fails to be a group since no element except 0 has an additive inverse within ℕ. Verifying all four axioms carefully distinguishes genuine groups from similar-looking structures.

Worked example

Verify the group axioms for the set of non-zero real numbers under multiplication, (ℝ\{0}, ×).

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  1. 1Closure: the product of two non-zero reals is non-zero, so closure holds.
  2. 2Associativity: multiplication of real numbers is associative by basic arithmetic.
  3. 3Identity: 1 satisfies 1·a = a·1 = a for all a ≠ 0, so 1 is the identity.
  4. 4Inverses: every a ≠ 0 has inverse 1/a with a·(1/a) = 1, so all axioms hold.

Answer: (ℝ\{0}, ×) is a group with identity 1

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Common mistake. Students assume any set with an operation that has an identity is automatically a group; (ℤ, ×) has identity 1 but most elements, like 2, lack a multiplicative inverse in ℤ, so it is not a group.

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