# Tagged Questions

201 views

### What are some properties that imply that a group must be the trivial group?

In the problem posed in this question of mine we want to show that a particular group is both perfect and solvable, and therefore trivial, and this turns out to be useful in proving the result. What ...
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### What are some examples of “exotic” algebraic structures? [closed]

I guess that I'm quite familiar with the basic "everyday algebraic structures" such as groups, rings, modules and algebras and Lie algebras. Of course, I also heard of magmas, semi-groups and monoids, ...
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### Differences between infinite-dimensional and finite-dimensional vector spaces

I've just started a course in Representation Theory, and in solving our first homework I've used a couple of theorems about finite-dimensional vector spaces (for an example, rank-nullity theorem). My ...
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### Interesting problems using group/representation theory

I've been going through this representation theory lecture notes, and I've found the ''hungry knights'' problem very interesting. Do you know any interesting problems similar to that one? To define ...
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### Examples of non-obvious isomorphisms following from the first isomorphism theorem

I am learning the first isomorphism theorem, and I am working with some isomorphisms to practice for my upcoming test. I know some of the basic ones like: $\mathbb{R}/\mathbb{Z} \cong \mathcal{C}$, ...
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### categorical generalizations of familiar objects

A couple of days ago I've learned that you can define trace in a very abstract setting. Namely, suppose $F\colon A\to B$ is a functor between two categories. Suppose $E,G\colon B\to A$ are two ...
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### Abstract Algebra/ Linear Algebra classic problems [closed]

I am studying for an Algebra/Linear Algebra exam coming up in August.In preparation for my exam I have worked on a lot of problems from Dummit and Foote and Hoffman and Kunze' books. I would like to ...
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### Which ring homomorphisms preserve/reflect what?

Exams are coming up and I'm getting kind of desperate. So more now than ever, whatever help you're able to provide is much appreciated. In the abstract algebra exam I'm currently preparing for, ...
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### Guides/tutorials to learn abstract algebra?

I recently read up a bit on symmetry groups and was interested by how they apply to even the Rubik's cube. I'm also intrigued by how group theory helps prove that "polynomials of degree $\gt4$ are not ...
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### Algebraic geometry in representation theory?

I heard that today algebraic geometry plays some significant role in representation theory, which is a little surprising because when I learnt representation theory it is basically algebra, topology, ...
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### Nice examples of groups which are not obviously groups

I am searching for some groups, where it is not so obvious that they are groups. In the lectures script there are only examples like $\mathbb{Z}$ under addition and other things like that. I ...
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### Worst category with first isomorphism?

I am no expert in category theory, but from VIII of Algebra: Chapter 0 I learnt that In an abelian category every $A\xrightarrow{\phi}B$ can be decomposed into A\twoheadrightarrow ...
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### Counterexamples in algebra

I got the feeling that whenever a subject gets so sophisticated that Zorn's lemma is needed, a book of counterexamples in that subject would probably benefit researchers/ students a lot. Zorn's ...
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### Examples of nonabelian groups.

Can anybody provide some examples of finite nonabelian groups which are not symmetric groups or dihedral groups?
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### Are there real world applications of finite group theory?

I would like to know whether there are examples where finite group theory can be directly applied to solve real world problems outside of mathematics. (Sufficiently applied mathematics such as ...
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### Applications of the Isomorphism theorems

In my study of groups, rings, modules etc, I've seen the three isomorphism theorems stated and proved many times. I use the first one ( $G/\ker \phi \cong \operatorname{im} \phi$ ) very often, but I ...
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### what is the most traditional abstract algebra textbook? and [Linear algebra & Abstract algebra] [closed]

I have listed 3 textbooks i have in my mind to buy Herstein - Topics in Algebra Artin - Algebra Lang - Undergraduate Algebra Unlike Lang's Algebra is the most traditional abstract algebra text for ...
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### Noetherian and Artinian rings (reference) [closed]

I started to study localization of rings and Noetherian and Artinian rings. Do you know any good references for these subjects? I'm using the one by Atiyah and Mcdonald. Is there another one? Thank ...
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### Examples of unusual group operations from outside of group theory.

Although it is certainly important to study frequently seen group operations like permutations, function composition, word operations, and so on, I find it fascinating to see group structure applied ...
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### Nice applications of the Haar measure

The existence of the Haar measure is a beautiful result that has a lot of applications. For example, one can prove using the Haar measure that the category of representations of a compact Lie group is ...
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### Concrete examples of abstract algebra delivering simple solutions to hard problems

Often people allude to examples of hard problems or theorems of which trivial solutions (or proofs) were found by applying techniques from abstract algebra. Can you give an example of such a problem ...
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### Applications of Abstract Algebra to elementary mathematics

I'm currently an undergraduate student in mathematics. I am currently taking Algebra. The course is interesting, but I have grown very curious about the usefulness of algebra. I am NOT asking about ...
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### Pathologies in module theory

Linear algebra is a very well-behaved part of mathematics. Soon after you have mastered the basics you got a good feeling for what kind of statements should be true -- even if you are not familiar ...
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### Cokernels - how to explain or get a good intuition of what they are or might be

When I think about kernels, I have many well-worked examples from group theory, rings and modules - in the earliest stages of dealing with abstract mathematical objects they seem to come up all over ...
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### Group theory applications along with a solved example

As I asked in previous question, I am very curious about applying Group theory. Still I have doubts about how I can apply group theory. I know about formal definitions and I can able to solve and ...
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### Real-world uses of Algebraic Structures

I am a Computer science student, and in discrete mathematics, I am learning about algebraic structures. In that I am having concepts like Group,semi-Groups etc... Previously I studied Graphs. I ...
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### Do there exist groups whose elements of finite order do not form a subgroup? [duplicate]

Possible Duplicate: Examples and further results about the order of the product of two elements in a group I was browsing around, and came across the little exercise that elements of finite ...
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### good books on Abstract Algebra and Cryptography for self-study

I want to self-study some abstract algebra and cryptography during the summer, so what are some of books that are suitable for self-study? I have very limited background in algebra and none in ...
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### Good introductory books on homological algebra

Which books would you recommend, for self-studying homological algebra, to a beginning graduate (or advanced undergraduate) student who has background in ring theory, modules, basic commutative ...