viii) If R is an integral domain, then R/I is an integral domain, for any ideal I of R. x) Every prime element of Z[x,X2,...,x,] is irreducible.
viii) If R is an integral domain, then R/I is an integral domain, for any ideal I of R. x) Every prime element of Z[x,X2,...,x,] is irreducible.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Please answer viii, ix and x
![1.
Which of the following statements are true? Give reasons for your answers. Marks will
only be given for valid justification of your answers.
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.
The number of distinct abelian groups of order p"p"
p, are distinct primes and n, e N.
iii)
* is n,n,...n,, where the
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) C0(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.
х)
|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%2F0139c7ad-758f-4a60-92d8-06e20ea7e80f%2Fxrr9gvd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1.
Which of the following statements are true? Give reasons for your answers. Marks will
only be given for valid justification of your answers.
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.
The number of distinct abelian groups of order p"p"
p, are distinct primes and n, e N.
iii)
* is n,n,...n,, where the
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) C0(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.
х)
|Aut (L/K)| = |Aut (L)| –|Aut(K).
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