MAT 511 HW #4 Name (JQEWUWK
10/4/13, due Friday 10/11/13
25 points
1. For each 0' 6 Sn given beEow, nd (a) the number of conjugates of (I in Sm {b} the order of the
centralizer Csnr), (c) the number of conjugates of the subgroup (a) in Sn, {(1) the ord
MAT 51} Exam 3
11/22/13 (due Wednesday 31/2Y/137 11:30 am)
160 points,-
Rules: You may C(msu'lt. your notes, our text and/or other books, and may discuss; the exam with mc but no other outsidm haip (inciuding internet)
is permitted. if you have questions
MAT 511
Exam 3
11/22/13 (due Wednesday 11/27/13, 11:30 am)
160 points
Name
Rules: You may consult your notes, our text and/or other books, and may discuss the exam with me, but no other outside help (including internet)
is permitted. If you have questions
MAT 511
8/30/13, due Friday 9/6/13
25 points
HW #1
Name
1. (Isaacs Problem 1) Let G be a group of functions from a set to itself, with the product dened
by composition of functions.
(a) Show, if G contains a one-to-one function, then G is a subgroup of th
MAT 511 Finai Exam Name
12/18/13
250 points
1.{25) Suppose f: G -> H and g: G > H are homomorphisms.
(a) Show that the set {at E G E ax) m is a subgroup of G.
f
u
2&5) (a) Prove: if G is a nite simple group, and (,0: G > G
MAT 511
9/7/13, due Friday 9/13/13
25 points
HW #2
Name
1. Let G be a group and g G. Let g : G G and g : G G be the functions dened by
g (x) = xg and g (x) = gx for x G. The functions : G SG , g g and : G SG , g g are
injective homomorphisms. Let R = im()
MAT 511
9/16/13, due Friday 9/20/13
25 points
HW #3
Name
1. Let G be a group.
(a) Prove that Inn(G) is a normal subgroup of Aut(G).
(b) Prove that Inn(G) is isomorphic to G/Z(G), where Z(G) is the center of G.
(c) Prove that Inn(G) cannot be cyclic, unles
MAT 511
10/4/13, due Friday 10/11/13
25 points
HW #4
Name
1. For each Sn given below, nd (a) the number of conjugates of in Sn , (b) the order of the
centralizer CSn (), (c) the number of conjugates of the subgroup in Sn , (d) the order of the
normalizer
MAT 511
Exam 2
10/25/13 (due Tuesday 10/29/13 at 6:00 pm)
150 points
Name
Rules: You may consult your notes, our text and/or other books, and may discuss the exam with me, but no other outside help (including internet)
is permitted. If you have questions,
MAT 511
Exam 1
9/27/13 (due Tuesday 10/1/13 at 6:00 pm)
140 points
Name
Rules: You may consult your notes, our text and/or other books, and may discuss the exam with me, but no other outside help (including
internet) is permitted. If you have questions, t
MAT 511
12/13/13
1.
HW #8 Solutions
(a) Let G be a nite group, and let V = C[G] be the group algebra of G over C, considered as a
right module over itself. Let v = gG g . Show that W = Cv is an irreducible submodule of V ,
and the corresponding representa
MAT 511
11/18/13, due Friday 11/22/13
25 points
HW #7
Name
1. Show any group of order 154 is solvable.
154 = 2 7 11. The number n11 of Sylow 11-subgroups divides 2 7 and is congruent to 1 mod 11, which
implies n11 = 1. Then the unique Sylow 11-subgroup P
m f:
MAT 511 HW #2 Name {SOIWM f if) PL
9/7/13, due Friday 9/13/13
25 points
if; 1. Let G be a. group and g E G. Let pg: G m) G and Ag: G w} G be the functions dened by
pg($) 2 mg and A903) 2 gm for :12 E G. The functions p: G + 53, g I> pg and A: G M}
MAT 511
10/13/13, due Friday 10/18/13
25 points
HW #5
Name
1. Use Burnsides Counting Formula to determine the number of dierent ways to color the four
edges of a square using n colors, as a function of n, where two colorings are considered to be same if
t