Please answer v, vi and vii 1.Which of the following statements are true? Give reasons for your answers. Marks willonly 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 witha 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 theiv)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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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. 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. 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, Ga group, and YCX is G-invariant, then X\Y is G-invariant.

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. 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. 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, Ga group, and YCX is G-invariant, then X\Y is G-invariant.

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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