(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)I = 10? (ii) Can |Z(G)|= 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,
(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)I = 10? (ii) Can |Z(G)|= 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,
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter4: More On Groups
Section4.4: Cosets Of A Subgroup
Problem 24E: Find two groups of order 6 that are not isomorphic.
Related questions
Question
![(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,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7944a9af-6eab-411c-849a-3f4a3337a38a%2Fa99994ca-c811-49db-94d6-84a0726d006c%2Fxgtjuwd_processed.png&w=3840&q=75)
Transcribed Image Text:(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,
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 3 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Recommended textbooks for you
![Elements Of Modern Algebra](https://www.bartleby.com/isbn_cover_images/9781285463230/9781285463230_smallCoverImage.gif)
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
![Elements Of Modern Algebra](https://www.bartleby.com/isbn_cover_images/9781285463230/9781285463230_smallCoverImage.gif)
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,