MA 4163/6163 - Homework















Starred* problems should be completed if you are enrolled in the course at the 6000 level.
I recommend (but do not require) that all students read and attempt these problems. If you make some significant progress, you should turn it in.

Homework 12, due never

Isaacs p81: 7.1, 7.2, 7.10

A. Explain why the quaternion group Q8 does not split nontrivially as a semidirect product.

B. Show how the symmetric group S4 splits as a semidirect product over its normal subgroup N of order 4. Be sure to describe the action of the complement H on N.

Homework 11, due Apr 27 at 12:00pm

Isaacs p81: 6.3, 6.4, 6.5, 6.7, 6.10, 6.12*

A. Show that a simple group of order 168 has subgroups of order 21, but no elements of order 21.

Homework 10, due Apr 15 at 12:00pm

Isaacs p67: 5.10, 5.13, 5.14, 5.18, 5.20*
Isaacs p81: 6.8

Homework 9, due Apr 1 at 12:00pm

Isaacs p67: 5.1, 5.2, 5.3, 5.7, 5.9, 5.12

Homework 8, due Mar 25 at 12:00pm

Isaacs p53: 4.3, 4.5

A. Suppose |G| = pa for a prime p and positive integer a. Show that Z(G) is nontrivial.

Hint: Considering fixed points under the conjugation action, and examine the class equation mod p.

B. The action of Sym [n] on [n] = {1, 2, ..., n} induces an action on subsets of [n], where {x1, x2, ...}˙σ = {x1˙σ, x2˙σ, ...}. Find the index and order of the stabilizer of a subset S of [n].

C. Let G be a finite group. Show that [H : HK] ≤ [G : K ], and that equality holds if and only if G = HK.

D. Let p be a prime, and G be a finite group. Show that if [G : H] = p, and P is a Sylow p-subgroup of G, then HP = G.

Hint: Problem C may be helpful for Problem D.

Homework 7, due Mar 18 at 12:00pm

Isaacs p39p53: 4.1, 4.4

A. Let G be a finite group and N be a normal subgroup contained in Φ( G ). Show that Φ( G/N ) = Φ( G ) / N.

B. Find a group G such that Φ( G ) is isomorphic to the cyclic group Z4, but show that there is no group K such that K/Φ( K ) is isomorphic to Z4.

C. Suppose that G is a group of order 60, and H is a subgroup of G having order 15. Show that G is not a simple group.

Note: There does exist a simple group S having order 60. By your argument for this problem, S has no subgroups of order 15.

D. Let V be an n-dimensional vector space over a field F with (a finite number) q elements. We previously calculated the order of G = GL(V). For any nonzero v in V, find the order of StabG v.

Homework 6, due Mar 4 at 12:00pm

Isaacs p39: 3.6, 3.8, 3.9, 3.10, 3.12

i. Show that if G is a permutation group on X, then G has a naturally defined group action on X.
ii. Explain why left conjugation ( Hg = gHg-1 ) does not define a group action on L(G) as defined in Isaacs, but right conjugation does.

Homework 5, due Feb 18 at 12:00pm

Isaacs p26: 2.9, 2.16
Isaacs p39: 3.1, 3.3, 3.4, 3.5*

A.  If H is any subgroup of G, prove that the subgroup generated by all conjugates of H is normal in G.
That is, prove that <Hg  :  g  G G

Homework 4, due Feb 11 at 12:00pm

Isaacs p26: 2.1, 2.6, 2.8, 2.10*, 2.17, 2.18, 2.21

Homework 3, due Feb 4 at 12:00pm

Isaacs p26: 2.3, 2.5, 2.7, 2.12, 2.13*, 2.19, 2.20

Homework 2, due Jan 28 at 12:00pm

Isaacs p12: 1.8
Isaacs p26: 2.2, 2.4

A. Prove that if H is a nonempty finite subset of a group G that is closed under multiplication, then H is a subgroup of G.

B. For each of the following subsets of GLn(F) (where F is any field), either prove or disprove that it forms a subgroup:
  i) The set of all matrices that are zero except on the diagonal.
  ii) The set of all matrices with determinant 1.
  iii) The set of all (weakly) upper triangular matrices.

C. Let m and n > 1 be relatively prime integers. Show that if G is an abelian group with elements of order m and of order n, then G has an element of order mn. Conclude that any abelian group of order 6 is cyclic.

Homework 1, due Jan 21 at 12:00pm

Isaacs p12: 1.1*, 1.2, 1.5, 1.6, 1.7, 1.9

A. For the triangular bipyramid obtained by gluing two regular tetrahedra together along a face, describe the rotation group, and the congruence group.