1)Let G = and H = {[0], [5]} Show that H is a normal subgroup of G List the elements of G/H and produce its

1)Let G = 10, +> and H = {[0], [5]}

Show that H is a normal subgroup of G

List the elements of G/H and produce its Cayley table

Show that G/H is isomorphic with Z₅

2)Let U = {Cos(x) + iSin(x) | x R}

Show that U is a group under normal multiplication

Prove the mapping f: -> defined by f(x) = cos(x) + isin(x) is a homomorphism

Prove Ker(f) = {2n | n Z}

Show that U R/ <2>

3) Consider the group Z^x₂₀ under multiplication

What are its elements?

which elements from a cyclic proper subgroup?

Find a non-cyclic proper subgroup ofZ^x₂₀