Question

Let H be a subgroup of a group G. Then H operates on G by left multiplication. Describe the orbits for this operation.

Answer 1

The orbits for the operation of the subgroup H on the group G by left multiplication are formed by multiplying each element of H with each element of G.

Step-by-step explanation:

In this case, the subgroup H operates on the group G by left multiplication. An orbit is the collection of elements in G that can be obtained by multiplying an element of H to an element of G. Let's say H = {h1, h2, h3} and G = {g1, g2, g3}. If we perform left multiplication of each hi with each gj, the orbits formed would be:

1. Orbit 1: {h1 * g1, h1 * g2, h1 * g3}
2. Orbit 2: {h2 * g1, h2 * g2, h2 * g3}
3. Orbit 3: {h3 * g1, h3 * g2, h3 * g3}