Homework Assignments
HW1. Due September 1, 2017
1. Show that subgroups of the additive group ℤ/mℤ are precisely the cyclic subgroups generated by all divisors of m.
2. Let G be the set of n x n upper triangular matrices with entries from ℤ/pℤ where p is a prime, and 1 on the diagonal. Show that G is a group (under matrix multiplication). Compute the number of elements in G and show that the order of each element in G is a power of p.
3. Every isomorphism from a group G to itself is called an automorphism. Show that the set Aut(G) of all automorphisms of G is a group under composition of maps and show that Aut(ℤ) is isomorphic to the 2-element group ℤ/2ℤ.
HW2. Due September 8, 2017
1. Let C be a cube inscribed in the sphere S. Show that the groupG of rotations of the sphere taking the cube to itself has 24 elements. Hint: first find 24 different rotations from G. Then consider the action of G on the set of large diagonals of the cube.
2. We color each square face of the cube C in 3 different colors. Two colorings are called equivalent if there exists a rotation of the cube that takes one coloring to another. Prove that there are 57 different colorings of the cube.
3. Show that if a group G has pn elements where p is prime, then it has non-identity a central element g, that is gh=hg for every h from G. Hint: Consider the action of G on G by conjugation.
HW3. Due September 15, 2017
1. Suppose that a group G has order 108. Show that it has a normal subgroup of order 27 or 9. Hint: Consider the action on the set of Sylow 3-subgroups by conjugation.
2. Supopose G is a group of order p2q where p, q are primes. Show that either a p-Sylow subgroup or a q-Sylow subgroup of G is normal.
3. In the group S4 let A be the subgroup generated by the cycle (1,2,3,4) and B be the subgroup generated by the cycle (1,2). Find all double cosets AgB.
4. Show that every group of order 30 contains only one 3-Sylow subgroup and one 5-Sylow subgroup.
HW4. Due September 22, 2017
1. Suppose that a cyclic subgroup T of a finite group G is normal in G. Show that every subgroup of T is normal in G as well.
2. Show that every group of order p2 (where p is prime ) is Abelian.
3. Let An be the group of even permutations of the set {1,2,...,n}. Let n> 4. Prove that An is generated by all 3-cycles (i,j,k) and also by all permutations (of the form (i,j)(k,l) where i,j,k,l are all different.
4. Show that every non-trivial normal subgroup of An, n> 4 must contain a 3-cycle or a permutation of the form (i,j)(k,l). Deduce that An, n>4 is simple. Hint: First prtove that An, n> 4 is the only normal subgroup of Sn apart from {1} and the whole Sn. Then use the fact that a minimal normal subgroup is the direct product of isomorphic simple groups.
HW 5. Due September 29, 2017
1. Show that every group of order 6 is isomorphic to either Z/6Z or to S3.
2. Let p,q be distinct prime numbers. Show that every group G of order pq is isomorphic to a semi-direct (possibly even direct) product of two cyclic groups of orders p and q. Deduce that G is solvable.
3. Suppose that a group G has normal subgroups K1, K2, ... and the intersection of all Ki is {1}. Show that G is isomorphic to a subgroup of the direct product of factor-groups G/Ki.
4. Show that every group of order p2, where p is prime, is either cyclic or isomorphic to the direct product of two cyclic groups of order p.
HW 6. Due October 6, 2017
All graphs here are in the sense of Serre.
1. Let G be the rewriting system where objects are all finite sequences of letters a, b and a positive move replaces two neighbor letters b,a by a,b. For example:
(b,b,b,a,a,b,a) → (b,b,b,a,a,a,b) → (b,b,a,b,a,a,b) → (b,a,b,b,a,a,b) → (b,a,b,a,b,a,b)
Prove that this rewriting system is Church-Rosser. Describe all terminal objects of this rewriting system.
2. Let G be the rewriting system where objects are natural numbers and positive moves are of two types. The moves of the first type take a perfect cube to its cube root. The moves of the second type takes a number n divisible by 3 to n/3. Show that this rewriting system is locally confluent and terminating and describe the terminal objects.
3. Let G be the rewriting system where objects are finite graphs and the positive moves are the following. If a graph has a closed path e1 e 2... ek (that is the end vertex of ek is the initial vertex of e1) without backtracts ee-1, then the move removes the edge ek from the graph. Is this rewriting system Church-Rosser?
HW 7. Due October 20, 2017 (because Friday 10/13 is a Fall break)
1. Exercise 1.8.22 from the book "Combinatorial algebra: syntax and semantics"
2. Exercise 1.8.23.
3. Exercise 1.8.24. Skip the last part of the exercise - about relatively free groups.
4. Exercise 1.8.26.
HW 8. Due Sunday, October 29, 2017
1. Find a connected 2-complex whose fundamental group is isomorphic to S3.
2. Show that the Baumslag-Solitar group BS(1,2) is isomorphic to the subgroup of the group of 2 x 2 matrices generated by the two matrices
and
3. Show that the group generated by the matrices
and
is isomorphic to the infinite dihedral group D∞.
HW 9. Due Sunday, November 5, 2017
1, 2. Exercise 5.8.1 (1) and (2) from my book.
3. Exercise 5.8.2.
HW 10. Due Sunday, November 12, 2017
1. Let H be the subgroup of the free group < a,b > generated by a2b-1, abba, aba-1. Does H contain a2? Does H contain a3?
2. Consider the linear representation R of S3 corresponding to the natural action of S3 on {1,2,3}. Find the alternating square of R.
3. Exercise 2.3 in Serre's book.
HW 11, Due Sunday, November 26, 2017
1. Problem 2.1 in Serre's book
2. Problem 2.2
3. Problem 2.4
4. Problem 2.5
5. Problem 2.6 (a)
6. Problem 2.7.