Homework for Week 8
Part 1
- Is the subgroup H normal in G? (Explain briefly.)
- H = 〈 2 〉 , G = Z/10Z
- H = 〈 (12)(3) 〉 , G = S3
- H = 〈 (123) 〉, G = S3
- For each normal subgroup in the previous problem, write the "multiplication" table of G/H (with respect to the group operation).
- Prove that if H is a normal subgroup of G then the map f given by f(g) = g * H is a group homomorphism from G to the quotient group G/H.
What is the kernel of the homomorphism?
- Prove that if G is an abelian group, then any quotient group of G is abelian.
- Prove that if G is a cyclic group, then any quotient group of G is cyclic.
- Prove or give a counterexample: If K is a normal subgroup of H, and if H is a normal subgroup of G, then is K a normal subgroup of G?
Part 2
- (Rotman 1.37) True or false, with reasons.
- If a finite group G acts on a set X then X must be finite.
- If a group acts on a finite set X then G must be finite.
- If a group G acts on a set X and if x,y are in X then Stab(x) is isomorphic to Stab(y).
- If a group G acts on a set X, and if x,y are both in the same orbit in X, then Stab(x) is isomorphic to Stab(y).
- (Rotman 1.24, parts) Try to relate this to group actions!
- How many permutations in S5 commute with (12)(34)?
- How many permutations in S7 commute with (12)(345)?
- List all of the permutations in S7 that commute with (12)(345)?
- How many rotational and reflectional symmetries of the pentagon are there?
- How many rotational symmetries does a cube have?
- (Recommended) The first problem from week 9: how many rotational symmetries does a dodecahedron have?
- Part 1: Due Tuesday 3/13
- Part 2: Due Friday 3/16