Abstract Algebra: A Study Guide for Beginners by Paul T. Bateman

By Paul T. Bateman

I first taught an summary algebra direction in 1968. utilizing Hcrstein's subject matters in Algebra. it really is challenging to enhance on his publication; the topic could have develop into broader, with purposes to computing and different parts, yet subject matters includes the middle of any path. regrettably, the topic hasn't turn into any more straightforward, so scholars assembly summary algebra nonetheless fight to profit the hot ideas, specifically considering that they're most likely nonetheless studying find out how to write their very own proofs.This "study consultant" is meant to aid scholars who're starting to find out about summary algebra. rather than simply increasing the fabric that's already written down in our textbook, i made a decision to attempt to coach by means of instance, via writing out strategies to difficulties. i have attempted to settle on difficulties that might be instructive, and in numerous situations i have integrated reviews to assist the reader see what's fairly happening. after all, this research consultant isn't really an alternative to an exceptional instructor, or for the opportunity to interact with different scholars on a few challenging problems.Finally. i need to gratefully recognize the help of Northern Illinois collage whereas scripting this learn advisor. As a part of the popularity as a "Presidential instructing Professor," i used to be given depart in Spring 2000 to paintings on initiatives concerning instructing.

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1 SOLUTIONS 20. The “Vertical Line Test” from calculus says that a curve in the xy-plane is the graph of a function of x if and only if no vertical line intersects the curve more than once. 1. Solution: We assume that the x-axis is the domain and the y-axis is the codomain of the function that is to be defined by the given curve. 1, a subset of the plane defines a function if for each element x in the domain there is a unique element y in the codomain such that (x, y) belongs to the subset of the plane.

Comment: Let’s use A for the matrix, and I for the identity matrix. The proof must be given in two pieces. We need to show that if 4|n, then An = I. 38 CHAPTER 1 SOLUTIONS We also need to show that An = I only when 4|n, and it is easier to state as the converse of the first statement: if An = I, then 4|n. The first half of the proof is easier than the second, since it just takes a computation. In the second half of the proof, if An = I then we will use the division algorithm, to divide n by 4, and then show that the remainder has to be 0.

1 5. Define the function φ : Z× , for all x ∈ Z× 17 → Z17 by φ(x) = x 17 . Is φ one to one? Is φ onto? If possible, find the inverse function φ−1 . −1 Solution: For all x ∈ Z× ) = (x−1 )−1 = x, so 17 we have φ(φ(x)) = φ(x −1 φ = φ , which also shows that φ is one-to-one and onto. 6. (a) Let α be a fixed element of Sn . Show that φα : Sn → Sn defined by φα (σ) = ασα−1 , for all σ ∈ Sn , is a one-to-one and onto function. Solution: If φα (σ) = φα (τ ), for σ, τ ∈ Sn , then ασα−1 = ατ α−1 . We can multiply on the left by α−1 and on the right by α, to get σ = τ , so φα is one-to-one.

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