Abstract Algebra

HW 3

February 26, 2018

https://penstrokeswriters.com/order/order

Due Friday, March 9. Be sure to justify all of your answers to receive full credit.

1. In this problem we will show that every group of order 15 is cyclic. Let G be a group of order 15.

The Sylow theorems (which we have not covered), say that there exist elements g of order 5 and h of order 3 in the group G.

(a) By considering the left cosets of 〈g〉, show that gh has the form hjgb, where 0 ≤ j ≤ 2 and 0 ≤ b ≤ 4.

(b) Show that gh = hgb, for some value of b which is not divisible by 5.

Hint: We may rule out j = 2 by showing that in that case gh has even order in G.

divisible by 3, but is not 3.

(d) Conclude that gh has order 15 and hence G is cyclic.

2. Saracino: 7.10, 8.11, 8.18, 9.6 (see pg 75 for notation), 9.18, 10.8, 10.15, 11.4, 11.16, 11.18.

SETS AND INDUCTION

One of the most fundamental notions in any part of mathematics is that of a set. You are probably already familiar with the basics about sets, but we will start out by running through them quickly, if for no other reason than to establish some notational conventions. After these generalities, we will make some remarks about the set of positive integers, and in particular about the method of mathematical induction, which will be useful to us in later proofs.

For us, a set will be just a collection of entities, called the elements or members of the set. We indicate that some object x is an element of a set S by writing xES. If x is not an element of S, we write x f£ S.

In order to specify a set S, we must indicate which objects are elements of S. If S is finite, we can do this by writing down all the elements inside braces. For example, we write

S={1,2,3,4}

to signify that S consists of the positive integers 1,2,3, and 4. If S is infinite, then we cannot list all its elements, but sometimes we can give enough of them to make it clear what set S is. For instance,

S= {1,4, 7,10,13,16, … }

indicates the set of all positive integers that are of the form 1 + 3k for some nonnegative integer k.

We can also specify a set by giving a criterion that determines which objects are in the set. Using this method, the set {l,2,3,4} could be denoted by

{xix is a positive integer ~4},

where the vertical bar stands for the words “such that.” Likewise, the set {1,4,7, 10, 13, 16, … } could be written as

{xix = 1 + 3k for some nonnegative integer k}.