(1) Let G be a group of order 94 = 2 x 47. Let Z(G), as usual, denote the center of G.(a) For each of the following, either just say "YES", or prove that the answer is always "No".(i) Can |Z(G)|= 10?(ii) Can |Z(G)I = 47?(iii) Can |Z(G)| = 2?(b) Must G have a subgroup of order 2? Why or why not?(c) If G is abelian, can G have more than 1 element of order 2? Why or why not?(d) Must G have a subgroup of order 47? Why or why not?For full credit, justify your answers carefully and completely, and use Lagrange's theorem if and when necessary,

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Answered: (1) Let G be a group of order 94 = 2 x…

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 23E: 23. Let be a group that has even order. Prove that there exists at least one element such that and...

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