Abstract Algebra Answers

(a) Let M and T be a groups and φ : M −→ T an epimorphism. Prove that if M is Abelian then T is Abelian.
(b) Let G be a group and let H be a subgroup of index 2. Prove that for every g ∈ G, g−1 / ∈ H whenever g / ∈ H. Hence, prove that for all a,b ∈ G, if a / ∈ H and b / ∈ H then ab ∈ H.
(c) Find all the distinct left cosets of the subgroup H = h(1,1)i in Z2 ×Z4.

What do you mean by fixed Matrix?

What is the total number of subgroup of the galois group of the polynomial x^3-2 over Q?

Show that the set of all real numbers of the form a+b√2 where 'a' and 'b' are integrals forms an integral domain

Let G be the set of all 3×3 non-singular matrices with entries from Q. Let X be a fixed matrix in G.prove that operation *defined by A*B= X^(-1)ABX be a binary operation. How do you prove that G is a group with respect to binary operation defined by A*B=X^(-1) ABX?

Give a non zero function from M3 (Z) to Z.

For the Ring, R= Z2[x]/<x^8-1>. Find zero divisors and nilpotent elements if any.

Using Fundamental theorem of homomorphism to prove that the rings R^2 and R^2/R^4 are isomorphic.

Give an example with justification of an element of M3 (Z) that is a unit but not the identity element.

Let G be a group and H be a non empty finite subset of G.If ab belongs to H for all a,b prove that H is a subgroup of G.will the result be true if H is not finite?