0
votes
0answers
61 views

Algebraic constructions to add a real to a sub-group of $\langle \mathbb{R}^+,.\rangle$

Consider the group $\langle \mathbb{R}^+,.\rangle$ and let $r$ be a real number (e.g. $r=\sqrt{2}$). I would like to know about all known algebraic constructions to build a sub-group of $\langle ...
4
votes
2answers
139 views

Different ways of constructing the free group over a set.

This could be too broad if we're not careful. I'm sorry if it ends up that way. Let's put together a list of different constructions of the free group $F_X$ over a given set $X$. It seems to be ...
3
votes
4answers
114 views

Examples of Infinite Simple Groups

I would like a list of infinite simple groups. I am only aware of $A_\infty$. Any example is welcome, but I'm particularly interested in examples of infinite fields and values of $n$ such that ...
7
votes
7answers
189 views

Examples of properties not preserved under homomorphism

An isomorphism indicates that two structures are the same, using different names for the elements. Therefore it's obvious that every (algebraic) property of the first structure must be present in the ...
5
votes
1answer
94 views

How do you compute group cohomology in practice?

If you have a finite group $G$ and a finite $G$-module $K$, and you need to know $H^1(G,K)$ or $H^2(G,K)$, how do you do it? Do you use a computer algebra system? (If so, which one?) Do you use a ...
6
votes
3answers
208 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 ...
7
votes
1answer
106 views

What is your favorite group? [closed]

I would like to know about your favorite group(s). Since groups do appear everywhere in mathematics and there are plenty of them, which ones have drawn your attention the most or surprised you? Please ...
0
votes
2answers
102 views

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 ...
4
votes
2answers
223 views

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: $\mathbb{R}/\mathbb{Z} \cong \mathcal{C}$, ...
4
votes
3answers
157 views

Alternative Creative Proofs that $A_4$ has no subgroups of order 6

Since I've been so immersed in group theory this semester, I have decided to focus on a certain curious fact: $A_4$ has no subgroups of order $6$. While I know how to prove this statement, I am ...
5
votes
0answers
87 views

Classes of groups known to be realizable (IGP)

A finite group $G$ of order $n$ is said to be realizable (over $\mathbb{Q}$) if there exists a Galois extension $L/\mathbb{Q}$ such that $\mathrm{Gal}(L/\mathbb{Q})=G$. I'm curious what classes of ...
7
votes
3answers
141 views

Realizing groups as symmetry groups

We're supposed to think of (non-Abelian) groups as groups of symmetries of some object. Sometimes it isn't obvious what this object is. For example, the fundamental group of a topological space acts ...
11
votes
5answers
291 views

Applications of Character Theory

Some of the applications of character theory are the proofs of Burnside $p^aq^b$ theorem, , Frobenius theorem and factorization of the group determinant (the problem which led Frobenius to character ...
152
votes
21answers
14k views

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 ...
30
votes
3answers
590 views

Alternative proofs that $A_5$ is simple

What different ways are there to prove that the group $A_5$ is simple? I've collected these so far: By directly working with the cycles: page 483 of ...
27
votes
9answers
5k views

Examples of nonabelian groups.

Can anybody provide some examples of finite nonabelian groups which are not symmetric groups or dihedral groups?
45
votes
8answers
4k views

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 ...
10
votes
7answers
1k views

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 ...
5
votes
2answers
185 views

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 ...
19
votes
1answer
232 views

For which categories we can solve $\text{Aut}(X) \cong G$ for every group $G$?

It is usually said that groups can (or should) be thought of as "symmetries of things". The reason is that the "things" which we study in mathematics usually form a category and for every object $X$ ...
20
votes
0answers
204 views

Projective profinite groups

I'm reading the first chapter of Serre's Galois Cohomology. On p. 58, He gives two examples of projective profinite groups: the profinite completion of free (discrete) groups; the cartesian product ...
17
votes
5answers
787 views

Elementary problems with group theoretic solutions

I am helping a friend develop a course in abstract algebra that is designed for high school students who have no knowledge of abstract algebra or any real exposure to formally rigorous mathematics. To ...
3
votes
1answer
357 views

Fundamental groups of certain 3-manifolds

I'm starting my master's thesis on geometry/topology & group theory. I'd like to know examples of fundamental groups of 3-manifolds having geometric structure of the following types: ...
6
votes
3answers
2k views

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 ...
5
votes
2answers
1k views

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 ...
9
votes
12answers
5k views

Introductory Group theory textbook

What textbook is good for introductory group theory?
25
votes
4answers
731 views

Simplicity of $A_n$

I have seen two proofs of the simplicity of $A_n,~ n \geq 5$ (Dummit & Foote, Hungerford). But, neither of them are such that they 'stick' to the head (at least my head). In a sense, I still do ...