Abstract Algebra Answers

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

1. (a) 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.

(b) Show that the set

P =



a2t

2 + a1t + a0 |a2 + a1 = a0 and a2, a1, a0 ∈ R



is a group under addition.

(c) Consider the set X = R\{−1} with the binary relation ∗ defined by x ∗ y = x + y + xy.

Find the solution for the equation 5 ∗ x ∗ 2 = −19.

Consider the set X = R \ {-1} with the binary relation * defined by x * y = x + y + xy.

Find the solution for the equation 5 * x * 2 = -19.

Show that the set P = {a₂t² + a₁t + a₀ |a₂+ a₁ = a₀ and a₂, a₁, a₀ ϵ R} is a group under addition

Any two groups of order m are isomorphic, where m ∈ N . True or False. justify

Compute the center of GL(3.F) Also find order of GL(3,3)

Define a relation R on the set of integers Z

by R= {(n, n+ 3k) I k belongs to Z}. Show that R is an equivalence relation. Also find all distinct

equivalence classes.

Let f be a degree two polynomial over R in n-variables and a = (a1, . . . , an) a

critical point of f. Show that there is a homogenous degree two polynomial Q

such that

f(x) − f(a) = Q(x − a) (x ∈ Rn

)