Abstract Algebra Answers

Using Cayley’s theorem, what is the permutation group to which a cyclic group of order 12 is isomorphic to?

Let G be a subgroup of GL₂ (Z₄) defined by the set {[m b ,0 1}] such that b ∈ Z₄ and m=±1. Show that G is isomorphic to a known group of order 8?

check whether or not Q[x]/<8x^3+6x^2-9x+24> is a feild or not

Find the cyclic subgroups of U(21).

Prove that (Z(sqrt(2)),+,×) is an integral domain.

Subject: Rings and fields

a) Using Cayley’s theorem, find the permutation group to which a cyclic group of

order 12 is isomorphic. (4)

b) Let τ be a fixed odd permutation in . S10 Show that every odd permutation in S is 10

a product of τ and some permutation in . A10 (2)

c) List two distinct cosets of < r > in , D10 where r is a reflection in . D10 (2)

d) Give the smallest n ∈ N for which An is non-abelian. Justify your answer.

1 Let f be a non trivial homomorphism from Z10 to Z15.Then which of the following holds?

A) im f is of order 10

B) ker f is of order 5

C) ker f is of order 2

D) f is a one to one map

2.the number of zeros of z⁵+3z²+1 in |z|<1,counted with multiplicity is

A)0 B) 1 C)2 D)3

How to solve this problems.

Which of the following is a zero divisor in the polynomial ring z12[x]. A)1+x, B)2+x, C)3+2x,D) 4+2x

Use the fundamental theorem of homomorphism for groups to prove the following theorem, which is called the zassenhaus

Use the fundamental theorem of homomorphism for groups to prove the following theorem, which is called the zassenhaus (butterfly) lemma