Math 210A: Algebra, Homework 4
Ian Coley
October 30, 2013
Problem 1.
Determine all conjugacy class in Sn for n 4.
Solution.
Henceforth, for the sake of notation when we must use variables, let k i denote the element
i + k in the symmetric group on n eleme
Math 210A: Algebra, Homework 2
Ian Coley
October 17, 2013
Problem 1.
Let G be a group and a, b G.
(a) Prove that an am = an+m and (an )m = anm .
(b) Prove that ord an =
ord a
(n,ord a)
if ord a < .
(c) Prove that ord ab = ord a ord b if a and b commute an
Math 210A: Algebra, Homework 1
Ian Coley
October 9, 2013
Problem 1. Let a1 , a2 , . . . , an be elements of a group G. Dene the product of the ai s by
induction: a1 a2 an = (a1 a2 an1 )an .
(a) Prove that
a1 a2 an b1 b2 bm = (a1 a2 an )(b1 b2 bm ).
(b) Pr
Math 210A: Algebra, Homework 3
Ian Coley
October 24, 2013
Problem 1.
Let H be a p-subgroup of a nite group G. Show that if H is not a Sylow p-subgroup, then
NG (H) = H.
Solution.
Since H is a p-group, it is contained in some Sylow p-subgroup P G. Since P
Math 210A: Algebra, Homework 6
Ian Coley
November 13, 2013
Problem 1.
For every two nonzero integers n and m construct an exact sequence
0 Z/nZ Z/mnZ Z/mZ 0.
For which n and m is the sequence split?
Solution.
Let : Z/nZ Z/mnZ be dened by (1) = m. Since 1
Math 210A: Algebra, Homework 7
Ian Coley
November 17, 2013
Problem 1.
(a) A morphism f : A B in a category C is called a monomorphism if for any two
morphisms g, h : C A, f g = f h implies g = h. Show that the composition of
two monomorphisms is a monomor
Math 210A: Algebra, Homework 8
Ian Coley
December 1, 2013
Problem 1.
Show that if 1 = 0 in a ring R, then R is the zero ring.
Solution.
Let x R be an element. Then x = 1 x = 0 x = 0. Therefore R is the zero ring.
Problem 2.
(a) For any ring R, dene a ring
Math 210A: Algebra, Homework 9
Ian Coley
December 3, 2013
Problem 1.
Determine all subrings of Z.
Solution.
Let R Z be a subring. Then we must have 1 R. Since R is closed under addition, we
have 1 + . . . + 1 = n R. Since R is closed under taking additive
Math 210A: Algebra, Homework 5
Ian Coley
November 5, 2013
Problem 1.
Prove that two elements and in Sn are conjugate if and only if type = type .
Solution.
Suppose rst that and are cycles. Suppose further that and are conjugate so
that = 1 . By earlier co
Lectures on Abstract Algebra
Preliminary Version
Richard Elman
Department of Mathematics,
University of California,
Los Angeles, CA 90095-1555, USA
Contents
Part 1. Preliminaries
1. Introduction
1
3
Chapter I. The Integers
2. Well-Ordering and Induction
3