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(4). Suppose that f: G→ G' is a monomorphism, G is a non-trivial group and G' is non-abelian. Then G can not be abelian. (5). If every subgroup of G is a finite group, then G must be a finite group. (6). Let Z14 acts on a set X with 3 elements. Then there are at least 2 orbits under the action.

(4). Suppose that f: G→ G' is a monomorphism, G is a non-trivial group and G' is non-abelian. Then G can not be abelian. (5). If every subgroup of G is a finite group, then G must be a finite group. (6). Let Z14 acts on a set X with 3 elements. Then there are at least 2 orbits under the action.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Determine whether the following statements are true or false. Justify your answer with brief explanations or counterexamples. (1). All groups of order 9 are isomorphic to each other. (2). The set of positive real numbers forms a monoid under multiplication. (3). If G is a group whose order is a prime number, then G is a simple group. Suppose that f: G → G' is a monomorphism, G is a non-trivial group and G' is non-abelian. Then G can not be abelian. (4). (5). (6). under the action. If every subgroup of G is a finite group, then G must be a finite group. Let Z₁4 acts on a set X with 3 elements. Then there are at least 2 orbits 14
Transcribed Image Text:Determine whether the following statements are true or false. Justify your answer with brief explanations or counterexamples. (1). All groups of order 9 are isomorphic to each other. (2). The set of positive real numbers forms a monoid under multiplication. (3). If G is a group whose order is a prime number, then G is a simple group. Suppose that f: G → G' is a monomorphism, G is a non-trivial group and G' is non-abelian. Then G can not be abelian. (4). (5). (6). under the action. If every subgroup of G is a finite group, then G must be a finite group. Let Z₁4 acts on a set X with 3 elements. Then there are at least 2 orbits 14
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