# Algebra [Lecture notes] by Mark Steinberger

By Mark Steinberger

Best elementary books

The universe in a handkerchief: Lewis Carroll's mathematical recreations, games, puzzles, and word plays

Consider you've gotten a material bag with one marble inside-either black or white, you do not know which. You upload a white marble, shake the bag, and take a marble at random. it is white. What are the chances that the remainder marble is white? evidently half, correct? incorrect. the proper resolution is 2/3. this can be only one of the rankings of interesting puzzles and paradoxes during this interesting booklet.

Express Review Guide: Basic Math and Pre-Algebra

Exhibit overview publications: simple arithmetic and Pre-Algebra is helping scholars gather useful talents with speedy, specified classes. All key issues are coated. Readers will reap the benefits of math information, ideas for heading off universal pitfalls, sidebars of math definitions, and an in depth word list. moreover, pre and posttests support scholars gauge either their vulnerable components in addition to their growth.

Confluence of Cosmology, Massive Neutrinos, Elementary Particles, and Gravitation

Justbefore the initial programof Orbis Scientiae 1998 went to press the scoop in physics used to be unexpectedly ruled through the invention that neutrinos are, in spite of everything, colossal debris. This was once anticipated via a few physicists together with Dr. Behram Kusunoglu, who had apaper released in this topic in 1976 within the actual evaluation.

Extra resources for Algebra [Lecture notes]

Example text

Show that a group G is cyclic if and only if there is a surjective homomorphism f : Z → G. † 8. Let f : G → G be a homomorphism. (a) Let H ⊂ G be a subgroup. Deﬁne f −1 (H ) ⊂ G by f −1 (H ) = {x ∈ G | f (x) ∈ H }. Show that f −1 (H ) is a subgroup of G which contains ker f . (b) Let H ⊂ G be a subgroup. Deﬁne f (H) ⊂ G by f (H) = {f (x) | x ∈ H}. Show that f (H) is a subgroup of G . If ker f ⊂ H, show that f −1 (f (H)) = H. (c) Suppose that H ⊂ im f . Show that f (f −1 (H )) = H . Deduce that there is a one-to-one correspondence between the subgroups of im f and those subgroups of G that contain ker f .

Let n be any power of 2. Show that every subgroup of Q4n other than e contains a2 . 10 Direct Products We give a way of constructing new groups from old. It satisﬁes an important universal property with respect to homomorphisms. 1. Let G and H be groups. Then the product (or direct product) of G and H is the group structure on the cartesian product G × H which is given by the binary operation (g1 , h1 ) · (g2 , h2 ) = (g1 g2 , h1 h2 ). We denote this group by G × H. 2. The direct product G × H is a group.

For m, m ∈ M , f (mm ) = f (m)f (m ). An isomorphism of monoids is a homomorphism f : M → M which is bijective. 21. 1. Show that any group with one element is isomorphic to the trivial group. 2. Show that any group with two elements is isomorphic to Z2 . 3. List the homomorphisms from Z9 to Z6 . 4. List the homomorphisms from Z5 to Z6 . 5. Show that the order of m in Zn is n/(m, n). Deduce that the order of each element divides the order of Zn . Deduce that every non-identity element of Zp has order p, for any prime p.