Abstract Algebra Answers

Show that the polynomial

X^5=9x+3 is not solvable by radicals over

Let X be a non-empty set. Define the set Z to be the collection of all subsets of X. Consider the binary operation “union” in the set Z, that is, for A and B that are subsets of X, let

A*B=A∪B


Determine which of the properties of the binary operation is/are satisfied in Z under *.

Let G=D_8, and let N={e,a^2,a^4,a^6}.
(a) Find all left cosets and all right cosets of N, and verify that N is a normal subgroup of G.
(b) Show that G/N has order 4, but is not cyclic.

Show that if g is a non cyclic group of order n then g has no elements of order n. Further give an example with justification of a non cyclic group all of whose proper subgroups are cyclic

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.

Let G be a group. Suppose that there are two elements a, b ϵ G with b ≠ ℮ satisfying

aba⁻¹ = b²,

where ℮ is the identity of the group. Prove that

a⁴ba⁻⁴ = bⁱ⁶

Determine whether or not the set W = {(x, y, z) ϵ Rᶟ xy = z} is a subgroup of the group

Rᶟ under coordinatewise addition.

Let α : Z₉* Z₂₇→ Z₂₇ be given by α ((a, b)) = 3b for a ϵ Z₉, b ϵ Z₂₇ and

(a, b) + (c, d) = (a + c, b + d) is given by addition modulo n for each group Zn.

i. Show that α is a homomorphism.

ii. Find Ker(α).

Let (G, *) be a group. Prove that the map π : G → G defined by π(g) = g * g is a

homomoprhism if and only if G is abelian