Modern Algebra I, Math 4613 and 5003, Section 001, Fall
2007
Homework 11, Due Dec 5: Sec 7.1, #7,11,14;Sec 7.2,
#3(b,c),12; Sec 7.3, #4,18,19.
Hint: #14(c), use factorization 1-(-x)^n=(1-(-x))(1+(-x)+…+(-x)^{n-1})
and show that none of the factors is 0 if x is nilpotent; 3(c) 2 powers series
are the same iff
the coefficients at the corresponding powers are
the same (work by definition);
3(c), to show that an inverse of a power series exists
look at the coefficients of the product.
#4, consider all possibilities for the image of 1; #18-19, work by
definition (an ideal is two-sided ideal), in 18(b) index collection by some
set.
Homework 10, Due Nov 28: Section 5.1, #1,4,14; 5.2, #1(a),2(a),4(b),5, 9(do
only 1st part with order 2 elements, 2nd follows from
8(b)).
Hint:#5,show that the order of (g_1,…,g_s) is lcm(|g_1|,…,|g_s|); #9, a
cyclic group has at most one subgroup of a given order.
Homework 9, Due Nov 19: Section 4.4, #2,3,12; Sec.
4.5, # 13, 20, 27; Sec 4.6, #2,5.
Extra: Sec 4.4, #1,13,14,15,17; Sec 4.5:
10,14,16,21
Hints: 4.4,#3:use the order of the elements;#12, consider the action on H
by conjugation and corresponding inclusion G/C_G(H) <Aut(H)
where you can use divisors of 3825=17*5^2*3^2; 4.5, #13, count n_7 and the number of elements of order 7 if n_7
is not 1;#20 similar to 13 but more counting; 4.6, #2: let H be normal in S_n and consider intersection H with A_n;
#5, let H be normal in G, consider decomposition of H into the union of (H
intersection G_i).
Exam 2 will be on Wednesday, Nov. 7.
Homework 8, Due Nov. 2: Section 4.3, #4,5,6,11(b,d), 13,19.
Extra: Sec 4.3, #8,10,12,20,21,22,25,27.
Alternative hint: #19, note C_G(x) is independent of x from K, but C_H(x)
is not; show C_H(gxg^{-1})=gC_H(x)g^{-1} (more
general than in #4).
Homework 7, Due October 26: Section 3.5, #3,4,15; Sect. 4.1, #4; Sec 4.2,
#4,8,9.
Extra: Sec. 3.5, 16; Sec 4.1, 10; 4.2, 11,12,13.
Homework 6, Due October 19: Section 3.2, #4,5,12,18;
Section 3.3, #3,4,9; Sec 3.4, #1.
Extra: Sec. 3.2, #8,10,11,19,20,21; Sec. 3.3,
#1,2.
Homework 5, Due October 5: Section 2.4, #19; Sec. 2.5, #13;
Sec. 3.1, #14(a,b,c),24,36,37.
Extra: Sec. 2.5, #14; Sec, 3.1, #1,3,15,16,22,23,30,31,38.
Exam 1 will be on Friday, September 28.
EXTRA SESSIONS on the extra problems on Wednesdays at 3:30
pm in MS502.
Homework 4, Due September 24: Sec.2.1,#7,8;Sec.2.2,#7,10;Sec.2.3,#1,19;Sec.2.4,#6.
(In Sec. 2.1, #7 groups are additive (+ operation), “power a^m” is written as m*a.)
Extra: Sec. 2.1, #2,5,9,10,11,13,12,15;Sec. 2.2, #1,2,3,5,6,9;Sec. 2.3,
#11,12,13,24;
Sec.2.4,#5,7,13,14.
Homework 3, Due September 14: Section 1.6, #2,6,7,17; Sec 1.7, #4(b),15,17.
Extra: Sec1.6, #1,3,4,12,13,16,18,20,22;Sec1.7, #6,11,12,18,19.
Homework 2, Due September 7: Section 1.3, #1,10,13; Sec 1.4:#2,8;Sec
1.5,#1.
Extra problems: Sec. 1.3, #3,6,7,11;Sec 1.4,#4,6,10.
Homework 1, Due August 26: Section 1.1, #5,9,12,15,20,25; Section 1.2,
#2,18;
Review Sections 0.1 and 0.2. Extra problems to prepare for exams (not for
submission): 1.1, #8,10,11,16,20,21,23,29; 1.2, #3,17.