Throughout in this text V will be a vector space of nite dimension n
over a eld K and T : V V will be a linear transformation.
1
Eigenvalues and Eigenvectors
A scalar K is an eigenvalue of T if there is a nonzero v V such that
T v = v. In this case v is c
1
Review
Last lecture we learnt how to orthogonally diagonalise a real symmetric matrix A.
Quadratic Forms: Conic Sections
2
The equation: ax 2 + bxy + cy 2 + dx + ey + f = 0 can always be
written in matrix form:
x
y
a
b/2
b/2
c
x
+d
y
e
x
+f =0
y
or as x
16
Notes on Jordan-Hlder
o
Denition 16.1. A normal series of a group G is a sequence of subgroups:
G = H0 H1 H2 Hm = 1
so that each subgroup is normal in the previous one (Hi Hi1 ). The quotient
groups Hi1 /Hi is called a subquotients. A renement of a nor
MATH 30A FALL 04
Homework This is homework from Math 30a, Fall 2004. I was teaching only one course. So I had a lot of time to prepare the course. The book was John Fraleigh A rst course in Abstract Algebra, 7th edition. We covered the rst four chapters.
4.3-QUS-12
S8:
An easy element of order 4 is (1234), so how to get others?
We can put two disjoint 4 cycles together: (1234)(5678)
We can also add a disjoint 2 cycles: (1234)(56), (1234)(56)(78)
That's it. Any cycle length divides the order of the el
(January 14, 2009)
[16.1] Let p be the smallest prime dividing the order of a nite group G. Show that a subgroup H of G of
index p is necessarily normal.
Let G act on cosets gH of H by left multiplication. This gives a homomorphism f of G to the group of
GROUPS: SUPPLEMENTARY TOPICS
Theorem (see Proposition 2.59, page 172):
If G is a nite abelian group, then G has a subgroup of order d for every divisor d of |G|.
Theorem (Cayley) (see Theorem 2.66, page 178):
Every group G is isomorphic to a subgroup of t
7.3.12. If G is a nite group of order n, and p is the least prime such
that p j n, show that any subgroup of index p is normal in G.
Let H be a subgroup of index p in G. Consider the action of G on the set
S of left cosets of H given by a xH = axH as in E