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Problem 1

Avg difficulty 3.0

Determine whether the following structures are groups. If they are, prove it. If not, explain why they are not groups.

  1. (Z,+)(\mathbb{Z},+).

  2. (Z∖{0},×)(\mathbb{Z}\setminus\{0\},\times) where ×\times denotes regular multiplication.

  3. (R∖{0},×)(\mathbb{R}\setminus\{0\},\times) where ×\times denotes regular multiplication.

  4. ({0},×)(\{0\},\times) where ×\times denotes regular multiplication.

  5. ({1},+)(\{1\},+) where ++ denotes regular addition.

  6. ({1},×)(\{1\},\times) where ×\times denotes regular multiplication.

Kan, A Complete Introduction to Abstract Algebra, Groups, Introduction To Groups, Problem 1

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Adapted from A Complete Introduction to Abstract Algebra © 2022–2026 Kan Onn Kit. Licensed under CC BY-SA 4.0. Source: A Complete Introduction to Abstract Algebra. Imported at revision a559e8b.