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

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

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.

## 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

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

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

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

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

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

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

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

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