My university offers two different two-semester sequences for learning abstract algebra, and I can't decide which one would be better for me, a physics major. Here are the two sequences and their course descriptions, copied and pasted from the university website:
Algebra 1: Theory of groups...
1. Homework Statement
Find all cosets of the subgroup H in the group G given below. What is the index (G : H)?
H = <(3,2,1)>, G = S3
2. Homework Equations
3. The Attempt at a Solution
I will leave out the initial (1,2,3) part of the permutation. We have S3 =...
<Moderator's note: Moved from a technical forum and thus no template. Also re-edited: Please use ## instead of $$.>
If ##R_{1}## and ##R_{2}## are relations on a set S with ##R_{1};R_{2}=I=R_{2};R_{1}##. Then ##R_{1}## and ##R_{2}## are bijective maps
##R_{1};R_{2}## is a composition of two...
This time my struggle is with ring ideals. Book still won't provide examples, so I'm again trying to come up with some of my own. I figured {0,2} might fit the definition as an ideal of ##\mathbb{Z/4Z}## since it is an additive subgroup and ##\forall x \in I, \forall r \in R: x\cdot r, r\cdot x...
So I'm just beginning to study abstract algebra and I'm not sure I grasp the definition of a quotient group, I believe it probably has to do with the book providing little to no examples. In trying to come up with my own examples, I imagined the following:
Consider the Klein four group, if we...
Hello! I just start looking at SDG and I'm already having difficulties with a few concepts as expressed by A Kock as:
"We denote the line, with its commutative ring structure (relative to some fixed choice of 0 and 1) by the letter R"
"The geometric line can, as soon as one chooses two...
I have been reading about Rings and Modules. I am trying reconcile my understanding with Lie groups.
Let G be a Matrix Lie group. The group acts on itself by left multiplication, i.e,
Lgh = gh where g,h ∈ G
Which corresponds to a translation by g.
Is this an example of a module over a ring...
Let ##M## be a left R-module and ##f:M \to M## an R-endomorphism.
Consider this infinite descending sequence of submodules of ##M##
##M \supseteq f(M) \supseteq f^2(M) \supseteq f^3(M) \supseteq \cdots (1)##
Can anybody show that the sequence (1) is strictly descending if ##f## is injective...
<Moderator's note: Moved from General Math to Differential Geometry.>
Let p:E→ B be a covering space with a group of Deck transformations Δ(p). Let b2 ∈ B be a basic point.
Suppose that the action of Δ(p) on p-1(b0) is transitive. Show that for all b ∈ B the action of Δ(p)on p-1(b) is also...
The group of moves for the 3x3x3 puzzle cube is the Rubik’s Cube group: https://en.wikipedia.org/wiki/Rubik%27s_Cube_group.
What are the groups of moves for NxNxN puzzle cubes called in general? Is there even a standardized term?
I've been trying to find literature on the groups for the...
1. Homework Statement
For the following sets, with the given binary operation, determine whether or not it forms a group, by checking the group axioms.
2. Homework Equations
(R,◦), where x◦y=2xy+1
(R*,◦), where x◦y=πxy and R* = R - {0}
3. The Attempt at a Solution
For question 1...
Hi, so I have just a small question about cyclic groups. Say I am trying to show that a group is cyclic. If I find that there is more than one element in that group that generates the whole group, is that fine? Essentially what I am asking is that can a cyclic group have more than one generator...
I am trying to learn about free groups(as part of my Bachelor's thesis), and was assigned with Hungerford's Algebra book. Unfortunately, the book uses some aspects from category theory(which I have not learned). If someone has an access to the book and can help me, I would be grateful.
First...
What does "linear" in linear algebra and "abstract" in abstract algebra stands for ?
Since I am learning linear algebra, I can guess why linear algebra is called so. In linear algebra, the introductory stuff is all related to solving systems of linear equations of form ##A\bf{X} = \bf{Y}##...
1. Homework Statement
Let ##A,B## be subgroups of a finite abelian group ##G##
Show that ##\langle g_1A \rangle \times \langle g_2A \rangle \cong \langle g_1,g_2 \rangle## where ##g_1,g_2 \in B## and ##A \cap B = \{e_G\}##
where ##g_1 A, g_2 A \in G/A## (which makes sense since ##G## is...
Hi,
I am a math undergraduate major and just finished my first abstract algebra course. Unfortunately, we used the lecturer's notes which are quite dry, without motivation, and it really felt bad. I am really interested in abstract algebra, and thus has decided to re-learn it over the summer...
The sum of the $k$ th power of n variables $\sum_{i=1}^{i=n} x_i^k$ is a symmetric polynomial, so it can be written as a sum of the elementary symmetric polynomials.
I do know about the Newton's identities, but just with the algorithm of proving the symmetric function theorem, what should we do...
1. Homework Statement
Prove that a finite group is the union of proper subgroups if and only if the group is not cyclic.
2. Homework Equations
None
3. The Attempt at a Solution
" => "
If the group, call it G, is a union of proper subgroups, then, for every subgroup, there is at least one...
Hello! I am a bit confused about the first Sylow theorem. So it says that if you have a group of order ##p^mn##, with gcd(n,p)=1, you must have a subgroup H of G of order ##p^m##. So, if I have a group G of order ##p^k##, there is only one subgroup of G of order ##p^k## which is G itself. Does...
1. Homework Statement
Note: I'm saying it's very very hard because I still couldn't solve it and I've posted it in stackexchange and no answer till now.
I'm posting here the problem statement, all variables and known data in addition to my solving attempts. Because I'm posting an image of my...
So I have two sets, call it ##A## and ##B##. I also have a function ##f:A\rightarrow B##. By themselves, it does not matter (or at the very least make sense) to think of ##A## and ##B## as, say, groups (I'm not really thinking exclusively about groups, just as an example). For that matter, it...
What are good books in universal algebra, given that I have a background in Herstein (Topics in Algebra), Hubbard/Hubbard, Engelking (Topology), and Dugundji (Topology)? I am currently reading Hungerford, and I found a field called universal algebra while searching internet for some concepts...
micromass submitted a new PF Insights post
How to self-study algebra. Part II: Abstract Algebra
https://www.physicsforums.com/insights/wp-content/uploads/2016/06/aastock6.png [Broken]
Continue reading the Original PF Insights Post.
What is the most motivating way to introduce group theory to first year undergraduate students? I am looking for some real life motivation or something which has a real impact.
micromass submitted a new PF Insights post
In High School and Want to Do Advanced Mathematics?
https://www.physicsforums.com/insights/wp-content/uploads/2016/03/highschool-math.png [Broken]
Continue reading the Original PF Insights Post.
1. Homework Statement
I am supposed to prove or disporve that ##f:\mathbb{R} \rightarrow \mathbb{R}##
##f(x)=\sqrt{x}## is onto. And prove or disprove that it is one to one
2. Homework Equations
3. The Attempt at a Solution
I know for certain that this function is not onto given the...
Hi. I have the following question:
Let G be a finite group. Let K be a subgroup of G and let N be a normal subgroup of G. Let P be a Sylow p-subgroup of K. Is PN/N is a Sylow p-subgroup of KN/N?
Here is what I think.
Since PN/N \cong P/(P \cap N), then PN/N is a p-subgroup of KN/N.
Now...
1. Homework Statement
Show that the group (\mathbb Z_4,_{+4}) is isomorphic to (\langle i\rangle,\cdot)?
2. Homework Equations
-Group isomorphism
3. The Attempt at a Solution
Let \mathbb Z_4=\{0,1,2,3\}.
(\mathbb Z_4,_{+4}) can be represented using Cayley's table:
\begin{array}{c|lcr}...
1. Homework Statement
For each ##n \in \mathbb{N}##, let ##A_{n}=\left\{n\right\}##. What are ##\bigcup_{n\in\mathbb{N}}A_{n}## and ##\bigcap_{n\in\mathbb{N}}A_{n}##.
2. Homework Equations
3. The Attempt at a Solution
I know that this involves natural numbers some how, I am just confused...