Abstract Algebra Answers

Let A=left−1,1,−i,iright,i=sqrt−1. Let the binary operation be multiplication, what is the inverse element of i?

a.\\(-i\\)

b.\\(i\\)

c.1

d.\\(-1\\)

Subtraction is an operation on Z which is neither commutative nor___

a.Distributive

b.Commutative

c.Additive

d.Associative

Every element

x in mathbbZ has ____________ inverses with to respect to addition as a binary operation.

a.\\(-x\\)

b.\\(0\\)

c.\\(frac{1}{x}\\)

d.\\(x\\)

Check whether or not S = { a0 + a1x + .... + an(x^n) belongs to Z[x] | 5 | a0} is an ideal of Z[x].

Is { [a a+b a+b b] | a,b belongs to Z} a subring of M2(Z)? Why, or why not?

Check whether any group of order 44 has a

proper normal subgroup or not.

1. Build up the operation tables for group G with orders 1, 2, 3 and 4 using the elements a, b, c, and e as the identity element in an appropriate way.

2. i. State the Lagrange’s theorem of group theory.

ii. For a subgroup H of a group G, prove the Lagrange’s theorem.

iii. Discuss whether a group H with order 6 can be a subgroup of a group with order 13 or not. Clearly state the reasons.

show that the roots of the equation z^4-1=0 form a cyclic group of order 4.

Prove that (R^5)/R ~_ R^4 as rings

p and Q are subgroups of a group G and o(P) and o(Q) relatively prime prove that p intersection Q=e,e belonge to G