Math 4720 & 5210 Fall 2010 -- Homework Problem Set #2

Due: Tuesday, September 21st
  1. Some random action questions.

    1. Suppose that \( D_{2 \,\cdot\, 4} = D_{8} \) (symmetries of a square) atcs on some set of 10 elements. Can this action be transitive? What are the possible sizes of orbits? [Explain your answers.]

    2. Draw a picture of a set of 8 points on which \( D_{8} \) acts transitively. Label your points and write down the corresponding permutation (determined by your action) for each element of \( D_8 \).

  2. Cylic problems.

    1. Find the subgroup lattice for \( \mathbb{Z}_4 \times \mathbb{Z}_3 \).

    2. Show that \( \mathbb{Z}_m \times \mathbb{Z}_n \) is cylic if and only if \( m \) and \( n \) are relatively prime.

  3. Cosets and Quotients

    1. Assume \(H \) is a subgroup of \( G \) of finite index and let \( H \leq K \leq G \). Prove that \( [G:H] = [G:K][K:H] \). Do NOT assume these groups are finite!

    2. 5210 Students: Let \( G \) be a finite group. Let \( N \) be a normal subgroup of \( G \) and let \( H \) be a subgroup (not necessarily normal). Show that if \( |H| \) and \( [G:N] \) are relatively prime, then \( H \) is a subgroup of \( N \).

      Hint: Consider image of \( H \) under the natural projection from \( G \) to \( G/N \).

  4. The Lattice Isomorphism Theorem (Section 3.3 page 99). Let \( G \) be a group with normal subgroup \( N \).
    Prove there is a bijection from the set of subgroups of \( G \) which contain \( N \) to the set of subgroups of \( G/N \).

    5210 Students: Prove at least 2 of the 5 properties of this bijection (see page 99).