COSETS AND LAGRANGES THEOREM
KEITH CONRAD
1. Introduction
Pick an integer m 6= 0. For a Z, the congruence class a mod m is the set of integers
a + mk as k runs over Z. We can write this set as a + mZ. This can be thought of as a
translated subgroup: start
2.6
Isomorphisms
Exercise 2.6.2. Describe all homomorphisms : Z+ Z+ . Determine which are
injective, which are surjective, and which are isomorphisms.
Solution. By the denition of homomorphism, for all positive n Z+ , we have
(n) = (1) + + (1),
(n) = (n),
Math 370 Homework 9 Fall 2009
(1) Let H, K be subgroups of G and let H K act on G by (h, k)g = hgk 1 .
(a) Show the (H K) double cosets of G are of the form HgK. (b) Show there
is a bijection between the H K double cosets and the H-orbits of G/K. (c)
Supp
Math 370 Homework 6, Fall 2009
(1) Let F be a eld and A M atmn (F ) with m < n. Show that there is
no B M atnm such that BA = In .
SOLUTION: Since the column dimension of A is greater than the row dimension, there is a nontrivial linear relation between t
Math 370 Homework 2, Fall 2009
(1a) Prove that every natural number N is congurent to the sum of its
decimal digits mod 9.
PROOF: Let the decimal representation of N be nd nd1 . . . n1 n0 so that
d
N = i=0 ni 10i . We want to show that this sum is congrue
Math 370 Homework 1, Fall 2009
(1) A polynomial f (x) with coecients in Q is a numerical polynomial if
f (n) Z for all n Z. Use induction to prove that for every natural number
k 0,
1
fk (x) := x(x 1) . . . (x k + 1)
k!
is a numerical polynomial. (f0 (x)
Math 370 Homework 7 Fall 2009
(1) Let A M atn (F) and f (x) = charpoly(A). Let TA : M atn (F)
M atn (F) be dened by TA (B) = AB. Find charpoly(TA ).
SOLUTION: The action of TA on B is simply the action of A on each column
of B, so TA can be thought of as
Math 370 Homework 3, Fall 2009
(1a) Is (Z/7Z) a cyclic group?
ANSWER: Yes, it is generated by (3 + 7Z) for example.
(1b) Is (Z/25Z) a cyclic group?
ANSWER: Yes, it is generated by (2 + 25Z) for example.
(1c) Is (Z/175Z) a cyclic group?
ANSWER: No. This on
Math 370 Homework 8 Fall 2009
(1) The group GL4 (F)GL4 (F) acts on M4 (F) by (C, D)A = CAD1 . Find
a set of representatives of the orbits of GL4 (F) GL4 (F).
SOLUTION: Using row-reductions, we can reduce any matrix to an uppertriagular matrix with only 1s
Math 370 Homework 10 Fall 2009
(1) Show Inn(G) Aut(G)
SOLUTION: Let Adx Inn(G) and Aut(G) be arbitrary. Consider the
action of the automorphism Adx 1 on arbitrary g G:
( Adx 1 )(g) = Adx (1 (g)
= (x1 (g)x1 )
= (x)(1 (g)(x1 )
= (x)g(x)1
= Ad(x) (g)
1
(2) L
Notes 9/28
Math 370
Recall: Last time, dened left (right) cosets
If H G and G =
H.
(aH), a disjoint union, then each left coset is in bijection with
when |G| < . Then G = ( H)[G : H], where the latter term is the number of (left)
H-cosets.
Example: (Z/pZ)
Math 370 class notes
Wednesday, November 25, 2009
page 1
Recall
G = 12
H 2 = 2-Sylow subgroup
H 3 = 3-Sylow subgroup
(1) if both are normal
Z / 4Z Z/3Z or
G H2 H 3 =
2
( Z / 2Z) Z/3Z
(2) H 2 G, not H 3
2
G A4 = cfw_e,. ( Z / 2Z)
( 3) H 3 G, not H 2
Z /
Exercise 2.4.3. Let a and b be elements of a group G. Prove that ab and ba have
the same order.
Proof. Suppose (ab)n = 1. We note that b = b(ab)n = (ba)n b, but this implies
(ba)n = 1, and so both have order n.
Exercise 2.4.6.
(a) Let G be a cyclic group
Math 370 Homework 2
Due: Week of Monday 9/19 in Lab
1. Left and right translation
Let G be a group and h G. Show that the map lh : G G, g 7 hg is bijective. Similarly, show that
the map rh : G G, g 7 gh is bijective.
2. The multiplication table
The multip
GROUP ACTIONS
KEITH CONRAD
1. Introduction
The symmetric groups Sn , alternating groups An , and (for n 3) dihedral groups Dn
behave, by their very definition, as permutations on certain sets. The groups Sn and An
both permute the set cfw_1, 2, . . . , n
TRANSITIVE GROUP ACTIONS
KEITH CONRAD
1. Introduction
Every action of a group on a set decomposes the set into orbits. The group acts on
each of the orbits and an orbit does not have sub-orbits (unequal orbits are disjoint), so
the decomposition of a set
HW2 Solutions
Section 16
13.) Let G be the additive group of real numbers. Let the action of G in the real
plane R2 be given by rotating the plane counterclockwise about the origin through
radians. Let P be a point other than the origin in the plane.
(a.
Math 370 Practice problems for the final
1. Let f : X Y and g : Y X be maps such that g f = idX . Show that g is surjective and that f is
injective.
2. Let M be an infinite set and let H = cfw_h S(M )| h(x) = x for all but finitely many x M .
(a) Show tha
This proves it is a homomorphism since all of the multiplications are accurate, and
is an isomorphism since every element in S3 is mapped to, with inverse dened by
matching entries. This shows f2 , f6 generate the group of functions since (12), (123)
gene
Cartesian (cross) product A B = cfw_(a, b) | a A, b B
Relation R from A to B, R A B
Function f : A B, a relation f A B with a !b (a, b) f
If (a, b) f we write f (a) = b
One-to-one (injective) function f if b 1 a with f (a) = b
Onto (surjective) function f
Comments on Problem Set 2
Math 250b
March 6, 2002
Here are the dimensions of the various Lie groups of Exercise 7.8 (problem 5 on this problem set). The
groups GLn R, SLn R, Bn , Nn , SOn R, and Sp2n R were done in class, but we give them here because som
Symmetry Groups and Pattern Types
Renee Scott
Northwest Missouri State University
Faculty Mentor: Mary Shepherd
Introduction
Patterns can be found everywhere in every aspect of life. They can be applied to
many subject areas and numerous occupations: from
2
2.1
Groups
Laws of Composition
Exercise 2.1.2. Prove the properties of inverses that are listed near the end of the
section.
Remark. The properties are listed on p. 40 as the following:
(a) If an element a has both a left inverse l and a right inverse r
Solution for (a). We claim that y = x1 w1 z. This follows since
x(x1 w1 z)z 1 w = xx1 w1 zz 1 w = 1.
Solution for (b). Suppose xyz = 1. This implies x1 = yz, and by Exercise 2.1.2(a),
this left inverse is a right inverse, and so 1 = xyz = x(yz) = (yz)x =
Excursion in elementary number theory
Notes for Math 370
Ching-Li Chai
1. Some facts about Z/nZ
(1.1) Let n 2 be a positive integer, and let
n = pe1 pea
1
a
be the primary factorization of n, where p1 , . . . , pr are distinct prime numbers, and e1 , . .
Examples of Group Actions
Notes for Math 370
Ching-Li Chai
In each of the following examples we will give a group G operating on a set S. We will
describe the orbit space G\S in each example, as well as some stabilizer subgroups StabG (x)
for elements x S
Math 370
Problem Set #7
Due week of Oct. 22, 2007, in lab.
Read Homan and Kunze, Chapter 3, Section 6.
1. From Homan and Kunze, Chapter 3, do these problems:
Pages 105-107, #4, 5, 11, 12, 17.
2. a) Let V, W, X be nite dimensional vector spaces, and let S
Math 370
Problem Set #6
Due week of Oct. 15, 2007, in lab.
Note: Due to Penns fall break, there will be no class on Monday, Oct. 15 and no lab on
Tuesday, Oct. 16. Students in the Tuesday lab may attend the Thursday lab instead if
they wish. They may also