Tagged Questions
7
votes
2answers
101 views
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, ...
1
vote
1answer
49 views
The relationship between inner automorphisms, commutativity, normality, and conjugacy.
An inner automorphism of a group $G$ is defined to be a function $f: G \to G$ such that for $x\in G$ $f(x) = a^{-1}xa.$
I have three somewhat broad questions about this:
Why is it related to ...
2
votes
1answer
90 views
Intuition on group actions
I'm trying to get more intuition on this definition:
Let $(G,\circ,e)$ be group.
A group action is a mapping $G×X→X:(g,x) ↦g.x \,$ such that:
\begin{align*}
∀x∈X &: e.x=x \tag{1}\\
∀g,h∈G,∀x∈X ...
5
votes
4answers
90 views
Intuitive Explanation of Morphism Theorem
Is there an intuitive explanation for the morphism theorem from introductory abstract algebra?
First Morphism Theorem: Let $K$ be the kernel of the group morphism $f: G \to H$. Then $G/K$ is ...
34
votes
4answers
834 views
Algebra: Best mental images
I'm curious how people think of Algebras (in the universal sense, i.e., monoids, groups, rings, etc.). Cayley diagrams of groups with few generators are useful for thinking about group actions on ...
9
votes
2answers
90 views
Intuition behind the Frattini subgroup
I am trying to get a better feel for what the Frattini subgroup really is, intuitively.
Let $G$ be a group and denote its Frattini subgroup by $\Phi(G)$. I know that $\Phi(G)$ is the intersection of ...
4
votes
0answers
50 views
Quotient-lifting properties
I borrowed this terminology from K. Conrad's article on series of subgroups, in which he discusses solvability of groups. This property of certain groups satisfies
Let $N\triangleleft G$. Then ...
1
vote
0answers
42 views
Duality between $[G,G]$ and $Z(G)$? [duplicate]
Possible Duplicate:
Center-commutator duality
Let $G$ be a group. It seems that there is a certain duality between two of its normal subgroups, the commutator \begin{equation}
...
3
votes
2answers
197 views
Why are only the first four alternating groups are non-simple?
I know asking for intuition in math is a generally flawed approach, but can anyone give any reason why only the first four alternating groups are non-simple?
2
votes
1answer
104 views
Intuitive interpretation of the adjacency matrix as a linear operator.
Naturally we can describe graphs via tables of "yes there is an edge" or "no there is not" between each pair of vertices, so the definition of an adjacency matrix is easily understood. Thinking of ...
10
votes
1answer
251 views
Centralizer, Normalizer and Stabilizer - intuition
What is the motivation/intuition behind these concepts? What notion/property of a group do they capture? Or what is the scenario of application.
Thanks.
9
votes
3answers
520 views
What's the Clifford algebra?
I'm reading a book on Clifford algebra for physicists. I don't quite understand it conceptually even if I can do most algebraic manipulations. Can some-one teach me what the Clifford algebra really ...
2
votes
1answer
114 views
What is the intuitive meaning of the adjugate matrix?
The definition of the adjugate matrix is easy to understand, but I have never seen it used for anything.
What is the intuitive meaning of this matrix?
Are there examples of applications which may ...
27
votes
0answers
1k views
Does a four-variable analog of the Hall-Witt identity exist?
Lately I have been thinking about commutator formulas, sparked by rereading the following paragraph in Isaacs (p.125):
An amazing commutator formula is the Hall-Witt identity: ...
6
votes
1answer
210 views
Understanding the three isomorphism theorems
I have learnt the following three isomorphisms for a while but without true understanding:
A group homomorphism $\phi:G\to G'$ can be decomposed into ...
5
votes
2answers
354 views
Intuition on the Orbit-Stabilizer Theorem
The Orbit-Stabilizer says that, given a group $G$ which acts on a set $X$, then there exists a bijection between the orbit of an element $x\in X$ and the set of left cosets of the stabilizer group of ...
9
votes
5answers
310 views
Intuition on group homomorphisms
So I'm studying for finals now, and came across the idea of homomorphisms again. This is not a new idea for me at all, having seen them in groups, rings, fields ect. However, on reevaluating them I ...
10
votes
4answers
297 views
What is the significance of multiplication (as distinct from addition) in algebra & ring theory?
In higher math, operators are defined over a set of objects; and these operators are usually denoted as addition and multiplication with a distribution rule. Assuming multiplication is not repeated ...
3
votes
2answers
126 views
What does Frattini length measure?
I have heard derived length, for example, described as a measure of "how non-commutative" the group is. An abelian group will have derived length $1$, whereas a non-solvable group will be so ...
8
votes
3answers
201 views
What does an outer automorphism look like?
I am working on a project in my group theory class to find an outer automorphism of $S_6$, which has already been addressed at length on this site and others. I have a prescription for how to go about ...
7
votes
2answers
182 views
Is there an idea of “primeless isomorphism” studied somewhere in finite group theory?
What I mean by "primeless isomorphism" is essentially a relation on finite groups by identifying groups whose structure differs only in which primes divide the groups' orders. The groups aren't ...
19
votes
3answers
504 views
Intuition behind Snake Lemma
I've been struggling with this for some time. I can prove the Snake Lemma, but I don't really “understand” it. By that I mean if no one told me Snake Lemma existed, I would not even ...
1
vote
1answer
170 views
What is the intuition behind the Fermat-Euler's Theorem?
Can someone give me an intuition behind the working of Fermat-Euler's theorem? I am not looking for definition nor for proof (I know both of them).
$$a^{\phi(p)} \equiv 1 \pmod p$$
This is what I ...
11
votes
2answers
242 views
Can we think of a chain homotopy as a homotopy?
I'm taking a course in algebraic topology, which includes an introduction to (simplicial) homology, and I'm looking for a bit of intuition regarding chain homotopies.
The definitions I'm using are:
...
30
votes
3answers
530 views
Why do we look at morphisms?
I am reading some lecture notes and in one paragraph there is the following motivation: "The best way to study spaces with a structure is usually to look at the maps between them preserving structure ...
40
votes
4answers
2k views
Why “characteristic zero” and not “infinite characteristic”?
The characteristic of a ring (with unity, say) is the smallest positive number $n$ such that $$\underbrace{1 + 1 + \cdots + 1}_{n \text{ times}} = 0,$$ provided such an $n$ exists. Otherwise, we ...
10
votes
3answers
722 views
Why are polynomials defined to be “formal”?
Despite the fact that $\forall n, n^3 + 2n \equiv 0 \pmod 3$, I understand that $n^3 + 2n$ (considered as a polynomial with coefficients in $\mathbb Z/3\mathbb Z$) is not equal to the zero polynomial.
...
1
vote
1answer
346 views
What is the intuition behind the proof of Abel-Ruffini theorem in abstract algebra?
Is there a way to explain this proof in Wikipedia without knowing the abstract algebra too much or deep function experience? In addition, I don't how the abstract algebra work even after I look at an ...
7
votes
3answers
398 views
Intuition regarding Chevalley-Warning Theorem
Three versions of the theorem are stated on pages 1-2 in these notes by Pete L. Clark:
http://math.uga.edu/~pete/4400ChevalleyWarning.pdf
Could anyone offer some intuitive way to think about this ...
5
votes
1answer
262 views
The only two rational values for cosine and their connection to the Kummer Rings
I am trying to understand Kummer Rings, and in particular what makes $n=3,4,6$ so special. (That is the Gaussian and Eisenstein integers)
The norm of $\mathbb{Z}[\zeta_n]$ is ...
10
votes
4answers
727 views
Why is it that Complex Numbers are algebraically closed?
I find it curious that Complex Numbers give enough flexibility to be algebraically closed, where the reals, rational numbers do not. For the reals it is easy to see that they cannot be used to solve ...
8
votes
5answers
944 views
Are cyclic groups always abelian?
If a group $C$ is cyclic, is it also abelian (commutative)? If so, is it possible to give an “easy” explanation of why this is?
Thanks in advance!
14
votes
2answers
442 views
Categorical description of algebraic structures
There is a well-known description of a group as "a category with one object in which all morphisms are invertible." As I understand it, the Yoneda Lemma applied to such a category is simply a ...
25
votes
4answers
1k views
Can someone explain the Yoneda Lemma to an applied mathematician?
I have trouble following the category-theoretic statement and proof of the Yoneda Lemma. Indeed, I followed a category theory course for 4-5 lectures (several years ago now) and felt like I understood ...
4
votes
4answers
860 views
Intuitive explanation of Nakayama's Lemma
Nakayama's lemma states that given a finitely generated $A$-module $M$, and $J(A)$ the Jacobson radical of $A$, with $I\subseteq J(A)$ some ideal, then if $IM=M$, we have $M=0$.
I've read the proof, ...
3
votes
1answer
435 views
Similarity between subtraction and division
I would like to hear some intuition about difference between subtraction and division. For binary subtraction operator the standard development is introduction of unary operation of taking negative ...
42
votes
8answers
2k views
Intuition in algebra?
My algebra background: I've had 2 undergrad semesters of algebra, a reading course in Galois Theory, a graduate course in commutative algebra and one in algebraic geometry, and I've done (most of) ...
11
votes
4answers
2k views
Importance of Cayley's theorem
I along with one of my friends were just discussing some basic things in group theory, when this question came up:
What are some fundamental results in group theory?
We happened to list out some:
...
1
vote
3answers
258 views
Intuition behind tensor expansions of linear maps
Given finite-dimensional vector spaces $V,W$, there is an isomorphism $\text{Hom}(V,W) \rightarrow V^* \otimes W$. In particular, any linear map $\phi : V \rightarrow W$ has a tensor expansion $\sum ...
6
votes
4answers
2k views
Finding all normal subgroups of a group
On my homework today, we had to find all the normal subgroups of $D_{n}$, the dihedral group of order 2n. I solved the problem by looking at how the conjugacy classes change based on whether n is even ...
