Which of the following statements are true? Give reasons for your answers. Marks will only be given for valid justification of your answers. 1. i) In a non-abelian group of order 27, the identity conjugacy class is the only class with a single element. ii) A finite field with 16 elements has a subfield with 8 elements. iii) The number of distinct abelian groups of order p"p...p is n,n,...n, where the P, are distinct primes and n, e N.
Which of the following statements are true? Give reasons for your answers. Marks will only be given for valid justification of your answers. 1. i) In a non-abelian group of order 27, the identity conjugacy class is the only class with a single element. ii) A finite field with 16 elements has a subfield with 8 elements. iii) The number of distinct abelian groups of order p"p...p is n,n,...n, where the P, are distinct primes and n, e N.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter3: Groups
Section3.4: Cyclic Groups
Problem 11E: Exercises
11. According to Exercise of section, if is prime, the nonzero elements of form a...
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Please answer as much questions as possible.
![Which of the following statements are true? Give reasons for your answers. Marks will
only be given for valid justification of your answers.
1.
i)
In a non-abelian group of order 27, the identity conjugacy class is the only class with
a single element.
ii)
A finite field with 16 elements has a subfield with 8 elements.
iii)
The number of distinct abelian groups of order p"'p"² ...pk is n,n,...n, where the
P; are distinct primes and
n;
e N.
iv)
If G is a finite group, such that Z(G) = G, then o(G) is a prime.
v)
If X is a G-set, G a group, and YcX is G-invariant, then X\Y is G-invariant.
vi) Any group of order 202 is simple.
vii) SL, (R) CO(3).
viii) If R is an integral domain, then R/I is an integral domain, for any ideal I of R.
ix) Every prime element of Z[x,, X,..,
X,] is irreducible.
|Aut (L/K)| = |Aut (L)|-|Aut(K).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff6989586-d674-4724-b04a-ded867deb35b%2F968e12f4-2bf3-4d51-9c66-38aaa79c78c7%2Fsof8pp5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Which of the following statements are true? Give reasons for your answers. Marks will
only be given for valid justification of your answers.
1.
i)
In a non-abelian group of order 27, the identity conjugacy class is the only class with
a single element.
ii)
A finite field with 16 elements has a subfield with 8 elements.
iii)
The number of distinct abelian groups of order p"'p"² ...pk is n,n,...n, where the
P; are distinct primes and
n;
e N.
iv)
If G is a finite group, such that Z(G) = G, then o(G) is a prime.
v)
If X is a G-set, G a group, and YcX is G-invariant, then X\Y is G-invariant.
vi) Any group of order 202 is simple.
vii) SL, (R) CO(3).
viii) If R is an integral domain, then R/I is an integral domain, for any ideal I of R.
ix) Every prime element of Z[x,, X,..,
X,] is irreducible.
|Aut (L/K)| = |Aut (L)|-|Aut(K).
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