Abstract Algebra Answers

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.

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

order 12 is isomorphic

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

Use the fundamental theorem ofhomomorphism for groups to prove

check whether or not q[x]/<8x³+6x²-9x+24> is a field.

If (a,b) E AxA, where A is a group, then o((a,b))=o(a)o(b)

Use the Fundamental Theorem of Homomorphism for Groups to prove the following theorem, which is called the Zassenhaus (Butterfly) Lemma:

Let H and K be subgroups of a group G and H′ and K′ be normal subgroups of H

and K, respectively. Then

i) H′(H ∩ K′) H′(H ∩ K)

ii) K′(H′∩ K) K′(H ∩ K)