Fall 2005 Math 411 Notes and Homework from Papantonopoulou's _Algebra:
Pure & Applied_ (due each MONDAY - the *ed problems are for you to
ponder - assignments are always subject to change as necessary):
9/12 Read Chapter 0, especially 0.3 on Z and mathematical induction.
Note: Papa uses |X| for what I called #X, the cardinality of a set X.
As you go through, try to do as many of the problems as you can,
especially:
0.1 # 3,4,6,7,8,10,13,15,16,18
0.2 # 1,2,3,5,12,13
0.3 # 1,5,10*,11*,14,15,16,17,24,31,32; also work out (by
induction) a formula for 1+3+5+...(2n-1)
0.4 # 1,8,11,15,19,25; also, compute (1+i)^8
0.5 # 4,7,10,15,17,18,20,21
9/19 1.1 # 3,5(cf.10),6,9,10(cf.5),12,13,18,21,23(for n=3,cf.12,13)
24, 26*,28*,29*
9/26 1.2 # 4,5,14,17,16,19,24,25,28,29,33,36
10/3 [Rosh Hashanah holidays 10/4-5, Ramadan begins 10/4]
[No class Monday 10/3, makeup Friday 10/7]
[HW due on Wednesday 10/5]
1.3 # 1,5,7,6,10,9,11*,13*,14*,15*,16,17,18*,19*,20,22*,24*
10/10 [Columbus Day 10/10, Yom Kippur holiday 10/13]
[HW due on Wednesday 10/12]
1.4 # 4,5,7,9,10,13,14,15,19*,20*,21*,35(we talked about this Friday)
10/17
1.4 # 16,18,23*,24*,25*,26,29,30,32,34,36,38,39,40,41*,42*,43*,44
[Please finish the 1.4 hw, all to be submitted Monday 10/24]
10/24
2.1 # 2,8,9,10,11,12,18,20,21,22,23*,26*,27*,24*,25*,28*,29,30,31,
32*,33*,35*,37*,34*,39
10/31
2.2 # 2,3,4,6,14,16*,17*,18,24,28,33,34,36,40,42,48*
11/7
2.3 # 2,6,7,8,12,13,14*,15*,16*,29*,20,21,24,26*,27*,28*
[The take-home midterm begins Wednesday 11/9 ...
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411 TAKE HOME MID-TERM
Work alone - you may use your book or other
published references, but you must cite these
as any responsible scholar would - it is best
if you work it out yourself, of course...!
1. Work out the groups (up to isomorphism) of low order, and exhibit
the various groups of each order using a group-multiplication table:
Prove that there is one group G of order 1, one with order 2, one with
order 3, two with order 4, one with order 5, two with order 6 and one
with order 7.
How many G are there of order 8 (can you prove that these are all)?
Indicate which of these G are abelian, and determine the centers Z(G)
of those which are not.
Now consider pairs of these groups G, G'(possibly G = G') and decide
whether or not there is a nontrivial homomorphism G -> G' (give an
example or prove there is none).
2. Work out and exhibit the subgroup graph of S_4, the group of all
permutations of 4 elements (or equivalently, the rotation symmetries
of a cube - the 4 "main diagonals" of the cube get permuted).
Indicate which subgroups are normal in the bigger groups in which they
are contained.
3. Let p be a prime, and let GL_n(Z/p) be the invertible n-by-n
matrices with entries in Z/p; in other words, the determinant of any
matrix in GL_n(Z/p) is invertible in Z/p, i.e. it is in U(p) = Z/p* =
{1,2,...,p-1}. Let SL_n(Z/p) be those matrices with determinant 1.
Show that SL_n(Z/p) is a normal subgroup of GL_n(Z/p) of index p-1,
and that the quotient group GL_n(Z/p)/SL_n(Z/p) is isomorphic to U(p)
= Z/p*.
What is the order of the groups GL_2(Z/3) and SL_2(Z/3)? Can you
figure out which groups these are isomorphic to? Can you find a
general formula for the order in the n-by-n case with any p?
4. Show that H = * = {1,-1,i,-i} is normal in G = Q_8 (the
quaternion group).
To what group is the quotient group G/H isomorphic?
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11/14 ... to be submitted Wednesday 11/16 - no class on Monday 11/14!]
11/21 [Thanksgiving holiday Thursday 11/24]
2.4 #3,10,16,19,22,27*,28*,29*,25*,26*,Cayley-(Di)Graphs problems*
2.5 #5,8,9,10,11,12,20*,21*,22*
11/28
3.1 #2,4,6,13,17,18*
3.2 #6,8,10,21*
3.3 #2,4,6,8,10*
12/5
3.4 #1,2,4,8,17,18
12/12
*****these are all optional - they aren't to be turned in*****
4.1 #1,5,6,7,10,11,12,13
4.2 #3,6,7,8,10,11
4.3 #1,3,8
4.4 Thm 4.4.4 (Class Equation) #1,8,10,11,12,13,14,15,16
4.5 Thm 4.5.4 (Conjugacy in S_n) #3,4,15
14
*****these are all optional - they aren't to be turned in*****
Review
[FINAL EXAM: 10:30 AM Saturday 12/17 LGRC A301
(the lowrise near science library)]
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Introduction to Abstract Algebra
Math 411 (Group Theory)
Final Exam
10:30 AM 17 December 2005
LGRC A301
Rob Kusner
1. Warm-ups
a. Suppose A and B are subgroups of C. If #A = 4 and #B = 9, then
#(A \cap B) = _____ and #C is at least _____?
b. If D is a group with #D = 29 then D \isom _____?
c. If \phi: E \to F is a homomorphisms between groups of order #E = 28
and #F = 27, what is ker(\phi)?
d. Determine all (non-isomorphic) abelian groups of order 900.
2. True/False
a. G \isom Z for every nontrivial G < Z.
b. If H < K are both abelian groups, then so is H/K.
c. If L/M is an abelian group, then so are L and M.
d. If N = N' \oplus N" and P = P' \oplus P", then N \times P =
(N' \times P') \oplus (N" \times P").
3. Symmetries
The most symmetric letter "X" (without serifs) has a symmetry group of
order 8, whereas adding a half-serif to one arm reduces the symmetries
to the trivial group, and adding a full-serif to one arm gives a
symmetry group of order 2. Explore the other decorations with full- or
half-serifs and the corresponding symmetry groups.
4. Hard-core
Cayley's Theorem implies that the cyclic group C_n can be realized as
a subgroup of the symmetric group S_n. Show that we can do a bit
better: for example, C_6 can be injected (1-to-1 homomorphism) into
S_5 (why? hint: 2+3=5), and C_30 into S_10 (why? hint: 2+3+5=10).
Find the smallest N(n) such that C_n injects into S_N(n).
5. Extra credit
Consider the "rational numbers mod the integers" - this is the abelian
group Q/Z under addition "mod 1"; for example 1/2 + 2/3 = 1/6 and 1/3
+ 1/3 + 1/3 = 0.
(i) Show that each g in Q/Z is of finite order.
Unfortunately, Q/Z not finite, nor is it even finitely generated, so
the fundamental theorem of finite (or finitely generated) abelian
groups fails to apply. Here is a remedy:
(ii) For each integer d > 1, let A(d) := {g = a/d^k}, that is, some
power of d is in the denominator. For example, A(2) = {0, 1/2, 1/4,
3/4, 1/8, 3/8, 5/8, 7/8, ... , a/2^k, ...}.
Show that:
(a) A(d) is a subgroup of Q/Z.
(b) A(c) \cap A(d) = {0} if gcd(c,d) = 1.
(c) Q/Z = \oplus_{p prime} A(p).
[Note A(p) is like an infinitely large version of Z/p^k: just as Z/p^k
contains all of the *smaller* p-power cyclic groups as subgroups, our
group A(p) contains *all* of the p-power cyclic groups!]
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