Abstract Algebra Answers

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

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.