Let (G, ○) be an arbitrary group with at least two elements. Which of the following statements is not true? (a) For every pair of elements a, b ∈ G, the element a ○ b also belongs to the set G. (b) For every element a ∈ G, there exists an element b ∈ G such that b ○ a = b. (c) If for some elements a, b, c ∈ G it holds that a ○ b = c, then b = c. (d) If e ∈ G is the neutral element, then for every element a ∈ G, there exists an element b ∈ G such that a ○ b = e. (e) If e ∈ G is the left-neutral element and f ∈ G is the right-neutral element, then e = f
Let (G, ○) be an arbitrary group with at least two elements. Which of the following statements is not true? (a) For every pair of elements a, b ∈ G, the element a ○ b also belongs to the set G. (b) For every element a ∈ G, there exists an element b ∈ G such that b ○ a = b. (c) If for some elements a, b, c ∈ G it holds that a ○ b = c, then b = c. (d) If e ∈ G is the neutral element, then for every element a ∈ G, there exists an element b ∈ G such that a ○ b = e. (e) If e ∈ G is the left-neutral element and f ∈ G is the right-neutral element, then e = f
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter3: Groups
Section3.2: Properties Of Group Elements
Problem 4E: An element x in a multiplicative group G is called idempotent if x2=x. Prove that the identity...
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Let (G, ○) be an arbitrary group with at least two elements. Which of the following statements is not true? (a) For every pair of elements a, b ∈ G, the element a ○ b also belongs to the set G. (b) For every element a ∈ G, there exists an element b ∈ G such that b ○ a = b. (c) If for some elements a, b, c ∈ G it holds that a ○ b = c, then b = c. (d) If e ∈ G is the neutral element, then for every element a ∈ G, there exists an element b ∈ G such that a ○ b = e. (e) If e ∈ G is the left-neutral element and f ∈ G is the right-neutral element, then e = f.
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