Math 3110 Spring 2010 -- Homework Problem Set #6

Due: Monday, April 19th
  1. Let \( H \) and \( K \) be subgroups of \( G \).

    1. Prove that \( H \cap K \) is a subgroup of \( G \).

    2. Suppose that \( H \) and \( K \) are normal subgroups (of \( G \)). Show that \( H \cap K \) is a normal subgroup of \( G \) as well.

    3. Let \( |G|=100 \), \( |H|=50 \), and \( |K|=20 \). Using Lagrange's Theorem, What are the possible orders of \( H \cap K \)?

  2. Let \( H = \{ 1,x^2,x^4 \} \subseteq D_6 = \{ 1,x,\dots,x^5,y,xy,\dots,x^5y \} \). It's not hard to see that \( H \) is a finite non-empty subset of \( D_6 \) and \( H \) is closed under the operation of \( D_6 \), so \( H \) is a subgroup of \( D_6 \). Quickly compute \( [D_6:H] \) (i.e. the index of \( H \) in \( D_6 \)). Then find all of the left and right cosets of \( H \) in \( D_6 \). Is \( H \) a normal subgroup of \( D_6 \)?

  3. Let \( G \) be a group. Recall that \( Z(G) = \{ x \in G \,|\, gx=xg \mbox{ for all } g \in G \} \) is called the center of \( G \). In a previous homework set we showed that \( Z(G) \) is a subgroup of \( G \).

    1. Quickly show that for \( a,b \in G \), \( ab=ba \Longleftrightarrow a = bab^{-1} \). Then prove that \( Z(G) \) is a normal subgroup of \( G \) (Just show \( Z(G) \) is normal, we already know it's a subgroup).

    2. Recall that \( Q = \{ \pm 1, \pm i, \pm j, \pm k \} \) is the quarternion group. It's easy to see that \( Z(Q) = \{ \pm 1 \} \), so \( \{ \pm 1 \} \) is a normal subgroup of \( Q \). Confirm that \( \{ \pm 1 \} \) is a normal subgroup of \( Q \) by showing all of it's left and right cosets match.

    3. Write down a Cayley table for \( Q / Z(Q) \). Is this quotient group abelian? Is this quotient group cylic? If so, find a generator. If not, show the orders of its elements are all too small, so that the quotient has no generator.

  4. Quotients in \( \mathbb{Z}_n \).

    1. Let \( H = \langle 5 \rangle \subseteq \mathbb{Z}_{100} \). First, \( H = \{ ??? \} \). Then compute the cosets of \( H \) in \( \mathbb{Z}_{100} \). Write down a Cayley table for \( \mathbb{Z}_{100} / H \). What familiar group is this isomorphic to?

    2. Let \( k, \ell, n \) be a positive integers such that \( n = k\ell \) (so \( k \) divides \( n \)). Make a conjecture about the what the quotient \( \mathbb{Z}_n / \langle k \rangle \) is isomorphic to. Then prove your conjecture. Hint: Define the map \( \varphi(x)=x \) from \( \mathbb{Z}_n \) to your target group and then use the first isomorphism theorem. Don't forget to prove that \( \varphi \) is a well-defined homomorphism.