PMATH 701 LECTURES
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4. Actions
Reference: Rotman, sections 2.7 and 5.2. Hungerford, chapter II sections 45.
Denition 4.1. An action of a group G on a set X is a function G X X,
denoted (g, x) g x, such that
(1) (gh) x = g (h x) for all g, h G and x X.
(2

PMATH 701 LECTURES
19
6. Structure of finite abelian groups
A group G is nitely generated if G = <X> for some nite set X G. (In
particular, every nite group is nitely generated.)
Theorem 6.1 (Fundamental Theorem of Finitely Generated Abelian Groups). Let

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PM 701
If N G, we write G/N for G/N . Note that the elements of G/N are the cosets
N a ( = aN ) of N . Also note that if G is nite, then |G/N | = [G : N ] = |G|/|N |.
subn
Proposition 2.6. If N H G and N
G, then N
H and H/N G/N .
Theorem 2.7 (Isomorphis

PMATH 701
Assignment 3
Fall 2009
Due Friday, November 6, 2009
1. (2006, #6) Let Z[ 5] be the ring Z[ 5] = cfw_a + b 5 : a, b Z.
(a) Prove that the element 2 is irreducible but not prime in Z[ 5].
(Hint: 2 3 = (a + 5)(1 5).)
(b) Prove that the ideal 2, 1 +

PMATH 701 PROJECT TOPICS FALL 2009
Each student taking this course for credit must complete a nal project, involving
a 20-minute oral presentation to the class and submission of a written report.
Presentations will be given in class during the rst week of

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PM 701
Corollary 4.7. If |G| = p2 with p prime, then G is abelian.
Proof. If G were nonabelian, then |Z(G)| = p by the previous corollary. Pick x
G\Z(G). Then Z(G) CG (x) = G, which forces Z(G) = CG (x) and hence x Z(G),
contradiction.
Corollary 4.8 (

PMATH 701 LECTURES
3
Theorem 1.4 (First Isomorphism Theorem). If : A B is a surjective homomorphism, then A/ B.
=
Proof. Dene : A/ B by [a] (a). Clearly well-dened and bijective.
Check preservation. Assume arity(f ) = n and [a1 ], . . . , [an ] A/ . Then

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PM 701
Suppose G is a nite group and H1 , . . . , Hk G with Hi G for 2 i k. We
claim that
(1) H1 H2 Hk G.
(2) |H1 H2 Hk | divides |Hi | |H2 | |Hk |.
(3) If gcd(|Hi |, |Hj |) = 1 for all i = j, then |H1 H2 Hk | = |Hi | |H2 | |Hk |.
Indeed, this is prove

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PM 701
7. Direct Products
Suppose G1 , G2 , . . . , Gn are groups and G := G1 G2 Gn . If H1 G1 ,
H2 G2 and so on, then the set H1 H2 Hn is always a subgroup of G.
(Note: in general, not every subgroup of G will have this form.) In the next few
paragrap

PMATH 701 LECTURES
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Example. G = S3 .
Let N be the unique Sylow 3-subgroup of S3 and K a Sylow 2-subgroup of S3 .
Concretely, N = cfw_id, (1 2 3), (1 3 2) while K is one of cfw_id, (1 2), cfw_id, (1 3) or
cfw_id, (2 3). Lets choose K = cfw_id, (1 2). Cl

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PM 701
Theorem 4.12 (Sylows Second Theorem). If G is a nite group and p is a prime
dividing |G|, then any two Sylow p-subgroups of G are conjugate.
Proof. Let H, K be Sylow p-subgroups of G, so |H| = |K| = pn and pn+1 does not
divide |G|. Consider the

PMATH 701
Due Friday, Oct. 9.
Assignment 1
Fall 2009
"Pure" Problems 1. (2007, #G1(c) Show that the group of inner automorphisms of a group G is a normal subgroup of the automorphism group, Aut(G), of G. 2. (2008, #3(a) Let p and q be distinct primes, p <

PMATH 701
Assignment 4
Fall 2009
Due Monday, December 14, 2009
1. (2000, #4(a) Give examples of rings R and R-modules M such that
(a) M is Artinian but not Noetherian. [Note: i.e., such that M satises the DCC on submodules
but not the ACC on submodules.]

PMATH 701
Due Friday, October 23, 2009
Assignment 2
Fall 2009
"Pure" Problems 1. (2008, #3(b) Let p and q be distinct primes, p < q. If p|(q - 1), show that there exists only one non-abelian group of order pq. 2. (2007, #G1(b) Prove that if G is a finite

PMATH 701 LECTURES
ROSS WILLARD
1. Algebraic Structures Reference: Stanley Burris, A Course in Universal Algebra, 1981, Chapter II, sections 12 and 56. Available for free download at http:/www.math.uwaterloo.ca/~snburris/htdocs/ualg.html. An algebraic typ