Abstract Algebra

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.

(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.

Abstract Algebra

If A⊂B, prove that (A⋃C)⊂(B⋃C) for any set C.

Abstract Algebra

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

Abstract Algebra

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.

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.

Abstract Algebra

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

aba⁻¹ = b²,

where ℮ is the identity of the group. Prove that

a⁴ba⁻⁴ = bⁱ⁶

Abstract Algebra

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

Rᶟ under coordinatewise addition.

Rᶟ under coordinatewise addition.

Abstract Algebra

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).

(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).

Abstract Algebra

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

homomoprhism if and only if G is abelian

Abstract Algebra

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.

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.

Abstract Algebra

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.

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