Abstract Algebra Answers

Let G be a group, H∆ G and β ≤ G/H . Let A = {x ∈G | Hx∈β} . Show that
i) A ≤ G ,
ii) H∆ A ,
iii) β = A/H .

For x ∈ G , define Hx = { g-¹ xg | g ∈ G } . Under what conditions on x will Hx ≤ G ?
Further, if Hx ≤ G , will Hx ∆ G ? Give reasons for your answer.

Give an example, with justification, of a group G with subgroups H and K such
that HK is not a subgroup of G .

Let G be a group and H ≤ G . Prove that the only right coset of H in G that is a
subgroup of G is H itself.

Write the Cayley tables for addition and multiplication in Z7.

Check whether or not { z ∈ C | z^5 = 1} is a group with respect to addition.

Let p be a prime and a ∈N such that p | a^50 . Show that p^50 | a^50.

Not every polynomial that is irreducible over Q[x] is irreducible over Z[x] . State whether the statement is true or false, justify with reason.

If ( R ,*1,*2) is a ring, then ψ : R × R → R :ψ(r1,r2) = r1 *2 r1 is a binary operation. State whether the given statement is true or false , justify with reason.

The function f ,defined by f(x) = (x-1)/(3-x) , has the same set as domain and as range. State whether the statement is true or false, justify the answer with reason.