Exercise 7.6. Show that each row and column of the group table contains all of the elements of G exactly once. Use this to show that there if |G| = 2 or 3, then there is only one possible group table. Later we can use this to deduce that there is exactly one group of order 2 and one group of order 3 up to isomorphism.
Exercise 7.6. Show that each row and column of the group table contains all of the elements of G exactly once. Use this to show that there if |G| = 2 or 3, then there is only one possible group table. Later we can use this to deduce that there is exactly one group of order 2 and one group of order 3 up to isomorphism.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter4: More On Groups
Section4.6: Quotient Groups
Problem 27E:
27. a. Show that a cyclic group of order has a cyclic group of order as a homomorphic image.
...
Related questions
Question
This is group theory question, please explain it using simple words step by step, thanks :)
Transcribed Image Text:Exercise 7.6. Show that each row and column of the group table contains all of the
elements of G exactly once.
Use this to show that there if |G| = 2 or 3, then there is only one possible group table.
Later we can use this to deduce that there is exactly one group of order 2 and one
group of order 3 up to isomorphism.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step 1: Given :
VIEWStep 2: Proof of the statement "Each element of G appears exactly once in each row and each column".
VIEWStep 3: Proof of the fact "There is only one possible group table if |G|= 2 or 3".
VIEWStep 4: Proof of "There is exactly one group of order 2 and one group of order 3 up to isomorphism".
VIEWSolution
VIEW
Step by step
Solved in 5 steps
Recommended textbooks for you
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,