10
votes
1answer
94 views

Geometric Intuition for Dihedral Group Automorphisms

I noticed the other day that the automorphism group of the dihedral group $D_{2n}$ (of order $2n$) is $\operatorname{Aff}(\mathbb Z/n\mathbb Z)$, the group of affine transformations of the $\mathbb ...
0
votes
1answer
26 views

Meaning of “Identify a set with another set” in group theory

There is a exercise problem that asks "Identify a set with another set ". I don't understand what I should do. Do I need to establish a bijection between them? Thanks EDIT-I: Actual question: G is a ...
0
votes
2answers
34 views

Finding/Recognising non-cyclic proper subgroups.

$Q$ is a multiplicative group of order $12$. You are given that two elements of $Q$ are $a$ and $r$ and that $r$ has order $6$ and $a^2=r^3$ You are also given that $a$ has order $4$, $a^2$ has order ...
1
vote
1answer
47 views

What is $R$-algebra and do I need to understand $R$-modules for it?

I was given the following definition of $R$-algebra: Let $R$ be a commutative ring. An $R$-algebra is a ring $A$ (with $1$) together with a ring homomorphism $f : R \to A$ such that ...
1
vote
1answer
51 views

Intuition - Normal Subgroup Test - Fraleigh p. 141 Theorem 14.13

(1.) Not querying proofs or formality. I do this in my other question. Normal Subgroup Test says H is normal in G $\iff gH{g}^{-1}\subseteq H$ for all $g \in G$. What's the intuition of this ...
3
votes
4answers
45 views

Equivalence Relation definitions of Coset - looks like 1-step Subgroup Test? [Fraleigh p. 97 theorem 10.1]

p. 4 We are especially interested in the case where the set is a group, and the equivalence relation has something to do with a given subgroup. That is, we want to partition a group G into subsets, ...
1
vote
1answer
98 views

What is the relationship between the second isomorphism theorem and the third one in group theory?

The second isomorphism theorem [wiki] in group theory is as follows: Let $G$ be a group. $H \triangleleft G, K \le G$. Then: $HK \le G$, $(H \cap K) \triangleleft K$, and $K/(H \cap K) ...
5
votes
2answers
112 views

What does the Yoneda lemma say for the identity functor and finite sets?

So I try to plug in the simplest arguments into the Yoneda lemma and see how to interpret it. I'll try it for the identity functor and the category of finite sets, in particular, I use an three ...
6
votes
2answers
85 views

Intuition - Fundamental Homomorphism Theorem - Fraleigh p. 139, 136

Let $\phi: G \to H$ be a group homomorphism with $K = \ker\phi$. Then $G/K \simeq \phi[G]. $ The hinge to the proof is to define $\Phi: G/K \to \phi[G]$ given by $\Phi(gK) = \phi(g)$. Then we must ...
6
votes
1answer
150 views

A subgroup has the same number of left and right cosets - Tricks - Fraleigh p. 103 10.32, 35

(32.) Let $H \le$ group G and let $a, b \in G.$ Prove or disprove. If ${aH= bH},$ then $Ha^{-1} = Hb^{-1}.$ $\color{blue}{Ha^{−1}} = \{\color{magenta}ha^{−1} | h ∈ H\} = ...
5
votes
2answers
97 views

Orthornomal matrices [duplicate]

Is there a more direct reason for the following: If the columns of $n\times n$ square matrix are orthonormal, then its rows are also orthonormal. The standard proof involves showing that left ...
11
votes
1answer
236 views

“Why” is $[\mathbb{C}:\mathbb{R}] < \infty$?

Obviously this question is a little open-ended. A lot of complex analysis seems to work primarily because we can view $\mathbb{C}$ as a finite-dimensional $\mathbb{R}$-algebra, and apply analytic and ...
5
votes
3answers
160 views

Intuition, proof, one-sided group definition - Any set with Associativity, Left Identity, Left Inverse is a Group - Fraleigh p.49 4.38

There's a similar post on this question but the third paragraph there is almost impossible to figure out. I explain in my profile why I have to use the big word "prognosticate." Proof that left ...
13
votes
1answer
213 views

Intuition about the class equation (and flowers).

I apologize in advance for the size of the images I've devoted a lot of time and effort to draw them on the computer and i didn’t manage to re-size. I'd be really thankful if anyone would edit this ...
4
votes
1answer
257 views

Intuition for Cayley Table and Cayley Table for identity, inverse but not associativity - Fraleigh p. 47 4.24

$1-2.$ I understand these proofs on pp. 5-6 for Cayley tables but what are the intuitions for Sudoku property : Every element of the group appears only once in each row and each column. Symmetric ...
8
votes
3answers
132 views

Intuition or Motivation behind definition of Homomorphism - Fraleigh p. 29

p.29: A binary algebraic structure is a set $S$ together with a binary operation $*$ on $S$ and is denoted $<S, *>$ p.29: Let $<S,*>$ and $<S',*'>$ be binary algebraic ...
0
votes
0answers
27 views

Understanding analytic construction of induced representation

I'd like to get some intuition for analytic construction of induced representations as described on Wikipedia. Algebraic construction also described there is much more intuitive and clear to me, but ...
1
vote
0answers
57 views

An $SU(3)$ isomorph in Clifford $G(5,0)$?

I am a computer scientist using the geometric (Clifford) algebras $G(n,0)$ over $\mathbb{Z}_3 = \{0,1,-1\}$ to describe distributed computations. My question concerns $G(5,0)$ with generators ...
1
vote
0answers
65 views

Understanding a.. weird definition

I came across the following definitons: Let $n$ be a positive integer and $\zeta_n$ be a primitive $n^{th}$ root of unity. For any prime $p\nmid n$, let the degree of the extension ...
3
votes
1answer
39 views

Why do the interesting antihomomorphisms tend to be involutions?

Given a semigroup $S$, define that an antihomomorphism on $S$ is a function $$* :S \rightarrow S$$ satisfying $(xy)^* = y^*x^*.$ Examples abound. Consider: Transposition, where $S$ equals the set ...
1
vote
1answer
86 views

Problem with structure of a semisimple ring theorem

Structure of semisimple ring (Wedderburn-Artin) in Rings and Categories of Modules - Frank W. Anderson, Kent R. Fuller (auth.) Proof: Please explain that: "Now $_RR$ is direct sum off these ...
3
votes
3answers
82 views

Counterexample for $A,B\triangleleft G,G/A\cong B \Rightarrow G/B\cong A$

Let $A,B\triangleleft G$. Give counterexample for the claims: a. $G/A\cong B \Rightarrow G/B\cong A$ b. $G/A\cong G/B\Leftrightarrow A\cong B$ I don't know from where to start. Can you ...
1
vote
2answers
59 views

Understanding concept of an operation being well defined for an equivalence relation

Let $I$ be an ideal in a ring $R$. Define the relation (congruence modulo $I$) by $a \equiv b$ if $b - a \in I$ Denotes the equivalence class containing $a$ by $\bar{a}$. Define $$\bar{a} + ...
9
votes
1answer
166 views

Could we reasonably classify finite groups?

So I have been reading some work on $p$-groups, and I noticed a particularly disturbing sentence: "There is no hope of finding a finite set of invariants that will define every p-group up to ...
7
votes
4answers
352 views

Visualising finite fields

I'm interested in finding visual and/or physical approaches to understanding finite fields. I know of a few: V. I. Arnold has a few pictures of 'finite circles' and 'finite tori' in his book Dynamics, ...
5
votes
1answer
229 views

The Degree of Zero Polynomial

I wonder why the degree of the zero polynomial is $-\infty$ ? I heard that, it is $-\infty$ to make the formula $\deg(fg)=\deg(f)+\deg(g)$ hold when one of these polynomials is zero. However, if that ...
1
vote
2answers
58 views

Clarification on Formal Definition of Functions

In my Abstract Algebra class, the professor defined a restriction as Given $ X\xrightarrow{f} Y $ and a non-void subset $S$ of $X$ define $ f \mid S\xrightarrow{S} Y $ by $(f \mid S )(s) = f(s), ...
1
vote
0answers
61 views

What does it mean for a coalgebra to be cogenerated by a subspace?

The usual definition of an algebra being generated by a subspace is as follows: Let $A$ be an algebra, $X \subset A$ a subspace, $\mathrm{Alg}(X)$ the free algebra generated by $X$. Then $A$ is ...
4
votes
0answers
165 views

What is the intuition of conjugacy classes?

How can I fully understand what are conjugacy classes are in groups? I know the definition, that $a$ and $b$ are conjugate if $gag^{-1}=b$ for some $g\in G$. But what is the intuition? Using a ...
6
votes
2answers
201 views

If $G$ is a finite group, $H$ is a subgroup of $G$, and $H\cong Z(G)$, can we conclude that $H=Z(G) $?

If $G$ is a finite group, $H$ is a subgroup of $G$, and $H\cong Z(G)$, can we conclude that $H=Z(G) $? I learnt that if two subgroups are isomorphic then it's not true that they act in the same ...
37
votes
8answers
2k views

Intuitive meaning of Exact Sequence

I'm currently learning about exact sequences in grad sch Algebra I course, but I really can't get the intuitive picture of the concept and why it is important at all. Can anyone explain them for me? ...
7
votes
2answers
330 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, ...
3
votes
1answer
171 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
180 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 ...
6
votes
4answers
292 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 ...
45
votes
4answers
2k 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 ...
11
votes
2answers
335 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
66 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
46 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
245 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?
4
votes
1answer
253 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 ...
11
votes
1answer
796 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.
12
votes
3answers
2k 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
266 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 ...
85
votes
2answers
3k 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: ...
7
votes
1answer
407 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 ...
8
votes
2answers
1k 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 ...
11
votes
5answers
820 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
400 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 ...
4
votes
2answers
200 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 ...