An introduction to abstract algebra, by F. M. Hall

By F. M. Hall

Show description

Read or Download An introduction to abstract algebra, PDF

Similar abstract books

Representation Theory and Complex Analysis: Lectures given at the C.I.M.E. Summer School held in Venice, Italy June 10–17, 2004

Six best specialists lecture on a large spectrum of modern effects with regards to the name, offering either a pretty good reference and deep insights on present study task. Michael Cowling offers a survey of assorted interactions among illustration idea and harmonic research on semisimple teams and symmetric areas.

Foundations of Analysis: A Straightforward Introduction: Book 2, Topological Ideas

This publication is an advent to the guidelines from common topology which are utilized in trouble-free research. it really is written at a degree that's meant to make the majority of the cloth obtainable to scholars within the latter a part of their first 12 months of analysis at a college or university even if scholars will more often than not meet lots of the paintings of their moment or later years.

Lie Groups and Algebraic Groups (Springer Series in Soviet Mathematics)

This can be a relatively remarkable booklet on Lie teams and algebraic teams. made out of hectographed notes in Russian from Moscow college, which for plenty of Soviet mathematicians were anything comparable to a "bible", the booklet has been considerably prolonged and arranged to strengthen the cloth during the posing of difficulties and to demonstrate it via a wealth of examples.

Towards the Mathematics of Quantum Field Theory, 1st Edition

This formidable and unique e-book units out to introduce to mathematicians (even together with graduate scholars ) the mathematical tools of theoretical and experimental quantum box conception, with an emphasis on coordinate-free shows of the mathematical items in use. This in flip promotes the interplay among mathematicians and physicists by means of providing a typical and versatile language for the nice of either groups, although mathematicians are the first objective.

Additional resources for An introduction to abstract algebra,

Sample text

Prove that G is abelian. 4–7 Let G1 and G2 be groups. Prove that G1 × G2 ∼ = G 2 × G1 . 4–8 Let G be a finite group and let x be an element of G. Prove that xn = e for some positive integer n. Hint: Use the fact that the elements x1 , x2 , x3 , . . cannot be distinct. 4–9 Let G be a group of even order. Prove that there exists a nonidentity element x of G such that x2 = e. 4–10 Let ϕ : G → G be an isomorphism from the group G to the group G . 1). Prove that ϕ−1 is an isomorphism. 4–11 Prove that Q is not isomorphic to Z (both groups under addition).

Let m and n be integers with n > 0. There exist unique integers q and r with 0 ≤ r < n such that m = qn + r. We omit the proof and instead appeal to the reader’s knowledge of long division, which is used to divide m by n as follows: 46 q n)m .. r m r =q+ n n ⇒ ⇒ m = qn + r. So the q in the theorem corresponds to the whole part of the quotient and the r corresponds to the remainder. In long division, one continues the algorithm until the remainder is less than the divisor, that is, 0 ≤ r < n (cf.

For x, y ∈ R, we have ϕ(x + y) = ex+y = ex ey = ϕ(x)ϕ(y), so ϕ satisfies the homomorphism property. Therefore, ϕ is an isomorphism and we conclude that R ∼ = R+ . 3 Example Is R isomorphic to Z? 6 that |R| = |Z|, that is, there is no bijection from R to Z. In particular, there cannot be an isomorphism from R to Z. Therefore, R is not isomorphic to Z. 35 If G ∼ = G , then G and G are indistinguishable as groups. If G has a property that can be described just using its elements and its binary operation, then G must have that same property, and vice versa.

Download PDF sample

Rated 4.33 of 5 – based on 7 votes