MA 4163/6163 - Homework
 

 

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Homework

Homework 12, due Apr 24 at 12:00pm

Please do the following problems from the textbook:

Isaacs p96: 7.5a, 7.10

and the following additional problems

A.  Describe A4 as a semidirect product. Make sure that you describe the action by automorphisms!

B.  Explain why the quaternion group Q cannot be decomposed as a semi-direct product.
Hint: examine the subgroup lattice!

Homework 11, due Apr 19 at 12:00pm

Please do the following problems from the textbook:

Isaacs p81: 6.8, 6.12
Isaacs p96: 7.1, 7.2, the n=2 case of 7.3

and the following additional problems

A.  Draw the subgroup lattice of Z5 x S3. Indicate normal subgroups with diamonds and non-normal subgroups with ovals.

B.  Draw the subgroup lattice of Z3 x S3. Indicate normal subgroups with diamonds and non-normal subgroups with ovals.
Your answers to A and B should be slightly different!

Homework 10, due Apr 12 at 12:00pm

Please do the following problems from the textbook:

Isaacs p81: 6.4, 6.5, 6.7, 6.13

and the following additional problems

A.  Let G = SL2( Z7 ) / ( Z( GL2( Z7) \cap SL2( Z7 ) ), where \cap denotes intersection.
     i. Verify that |G| = 168.
     ii. Exhibit at least 2 subgroups of G with order 3, and at least 2 subgroups of G with order 7. (It may be helpful to first find these subgroups in SL2, and show that they are preserved in the quotient.)

B.  Let G be as in the previous problem. By following the proof technique used to show that A5 is simple, show that G is a simple group if G has a proper normal subgroup, then it has a normal 2-subgroup. You may use without proof the fact that G does not embed injectively into S7 have 7 Sylow p-subgroups.

C.  Describe the isomorphism type of a Sylow 2-subgroup of A5. Describe the isomorphism types of a Sylow 2-subgroup and of a Sylow 3-subgroup of A6.

Homework 9, due Apr 5 at 12:00pm

Please do the following problems from the textbook:

Isaacs p67: 5.8, 5.13

and the following additional problems

A.  Prove that any group of order 7735 is cyclic.

B.  Show that S8 contains an element of order 15, but that S7 contains no such element.

C.  Find all Sylow 2- and 3-subgroups of S4. Include a discussion of how many such subgroups there are, and the isomorphism type of each such subgroup.

D.  Find all Sylow 2-, 3-, and 5-subgroups of S5. Include a discussion of how many such subgroups there are, and the isomorphism type of each such subgroup.

Homework 8, due Mar 22 at 12:00pm

Please do the following problems from the textbook:

Isaacs p67: 5.7, 5.9, 5.10, 5.11, 5.14, 5.16

Homework 7, due Mar 8 at 12:00pm

Please do the following problems from the textbook:

Isaacs p53: 4.4
Isaacs p67: 5.1, 5.2, 5.3, 5.18

and the following additional problems

A.  Describe up to isomorphism all groups of order pq, where p > q are primes such that q does not divide p - 1.

Hint: On 5.3, use the Sylow D Theorem. Intersecting H with Sylow subgroups of G may be helpful.
Hint: On 5.18, Corollary 4.5 may be helpful.

Homework 6, due Mar 1 at 12:00pm

Please do the following problems from the textbook:

Isaacs p53: 4.1, 4.2, 4.3, 4.5

and the following additional problems

A.  Explain why left multiplication doesn't give a (right) group action: i.e., explain why xg = gx does not define a group action.

B.  Prove that if G is a finite group of order pk, then the center Z(G) is nontrivial.
Hint: consider the class equation. How big is the conjugacy class of an element of Z?

C.  Recall that the normal closure of a group H is the subgroup NC(H) generated by all conjugates of H (i.e., NC(H) = <Hg : g in G>). Prove that if N is normal in G, and H is contained in N, then NC(H) is contained in N.

Homework 5, due Feb 25 at 12:00pm

Please do the following problems from the textbook:

Isaacs p39: 3.4, 3.5, 3.8, 3.10, 3.12, 3.13, 3.14

and the following additional problems

A.  Describe up to isomorphism all groups of order 2p, where p > 2 is a prime. (We discussed p = 3 in class.)

B.  Recall that the Frattini subgroup of G, is the intersection of all maximal subgroups of G, and is written Phi(G) (with a capital Greek letter). Show that Phi( G / Phi(G)) is the trivial subgroup.

Remark: On 3.4b, H should be taken to be a Hall pi-group.

Homework 4, due Feb 8 at 12:00pm

Please do the following problems from the textbook:

Isaacs p26: 2.16, 2.18, 2.21
Isaacs p39: 3.1, 3.3, 3.9

and the following additional problems

A.  Prove Lemma 3.5 part b. (I.e., work out all of the routine details!)

B.  Show that every group of order 6 is isomorphic to either Z6 or S3.
Hint: Find the order of the subgroup generated by every element, and by every pair of elements. It may be helpful to divide into abelian and nonabelian cases.

Homework 3, due Feb 1 at 12:00pm

Please do the following problems from the textbook:

Isaacs p26: 2.3, 2.8, 2.11, 2.12ab, 2.13, 2.20

and the following additional problems

A.  Show that every subgroup of the cyclic group Zn is characteristic.

B.  If H is a subgroup of G, prove that the intersection of all conjugates Hg (over all g in G) is normal in G.

Homework 2, due Jan 25 at 12:00pm

If you didn't come to office hours last week due to snow, and come this week instead, you can still get 2 bonus points.

Please do the following problems from the textbook:

Isaacs p26: 2.1, 2.2, 2.4, 2.5, 2.6, 2.7

and the following additional problems

A.  Work out the details of the following:
a) Show that conjugation by any g in G is an automorphism of G
b) Show that if H is a subgroup of G and f is an isomorphism from G to K, then f(H) is a subgroup of K.
c) Show that for any group G we have Aut(G) to be a group.
d) Find the group structure of Aut(Z12).

B.  Find the subgroup lattice of Q8 and D8. Indicate the normal subgroups with diamonds, and the non-normal subgroups with circles or ovals.

C.  Construct an abelian group A with 8 elements such that A has an element of order 4, but no element of order 8.

Hint on 2.5: Relate elements of G minus M to the subgroup lattice.
Note on 2.6: You may assume G is finite -- in this case, |G:H| = |G| / |H|. This problem will be easier if you read ahead and use (2.19), (2.20), (2.21), and/or (2.22), which you should feel free to do.

Homework 1, due Jan 18 at 12:00pm

Please do the following problems from the textbook:

Isaacs p12: 1.1, 1.2, 1.5, 1.6, 1.7, 1.8, 1.9

and the following additional problems

A.  We defined D2n to consist of all rotations and flips of the n-gon. Prove that D2n is a permutation group.

B.  In problem 1.7, notice that GL(2,2) has order 6. Is this group isomorphic to the integers mod 6? To Sym( {1,2,3} )? To neither?
(Take the group operations to be multiplication on GL, addition on Z/6Z, and composition on Sym.)

C.  On this homework set (only), you can get 2 bonus points for coming by my office hours on Thursday with a question related to the course material.