0
votes
1answer
49 views

What would be an example such that $aH=bH$ but $Ha \neq Hb$?

Let $H$ be a subgroup of a group $G$. Assume $aH=bH$ for some $a,b\in G$ What would be an example such that $Hb\neq Ha$? I cannot imagine what would be.. Moreover, if $H$ is infinite, ...
6
votes
1answer
60 views

Inner Product on a Vector Space over a field besides $\mathbb R$ or $\mathbb C$?

Are there any fields with vector spaces you can define an inner product over besides subfields of $\mathbb C$? I know that you'd want the field to contain an ordered subfield, so it must have ...
10
votes
1answer
107 views

If the tensor power $M^{\otimes n} = 0$, is it possible that $M^{\otimes n-1}$ is nonzero?

Let $M$ be a module over a commutative ring $R$. It is possible that $M \otimes M = 0$ if $M$ is nonzero, for example when $R = \mathbb{Z}$ and $M = \mathbb{Q}/ \mathbb{Z}$. What about when higher ...
2
votes
1answer
51 views

A question about polynomials in $K[x_1,x_2,…,x_n]$ and there permutations

Let $K$ be a field. Let $n$ be a positive integer and $P$ be a non-symmetric polynomial in $K[x_1,x_2,...,x_n]$. $S_n$ acts on $K[x_1,...,x_n]$ in an obvious way. Let $P_1,P_2,...,P_r $ be the ...
1
vote
0answers
45 views

Heineken and Mohamed Groups

I was trying to understand the costruction of the Heineken and Mohamed groups. (II) Every proper subgroup of $G$ is subnormal and nilpotent. Lemma 1. If $G$ is a non-nilpotent soluble group ...
8
votes
2answers
97 views

Can we make $\mathbb{Z}$ into a field?

This is probably an elementary question about fields, but I think it is a little tricky. Can we make the integers $\mathbb{Z}$ into a field? Let me be more precise. Is it possible to make ...
7
votes
1answer
80 views

Localization of an additive category which is no longer additive

Is there a nice example of an additive category $C$ and a family of morphisms $S\subset Mor(C)$ such that $C[S^{-1}]$ is no longer additive? I know that in general localization of categories ...
4
votes
1answer
73 views

Weakened associativity axiom: $(x * y) * z \leq x*(y*z).$

Call a partially ordered magma $(X,*)$ sub-associative iff it satisfies the following axiom. $$(x*y)*z \leq x*(y*z).$$ Basically, this is saying that we may shuffle brackets right to get a larger ...
1
vote
3answers
134 views

Counterexamples to Nakayama's Lemma if $M$ is not finitely generated

One of the most famous forms of Nakayama's lemma says: Let $I$ be an ideal in $R$ and $M$ a finitely-generated $R$ module. If $IM = M$, then there exists an $r \in R$ with $r ≡ 1 \pmod I$, ...
2
votes
3answers
168 views

Counterexample to linear transformation.

What counterexample can I use to prove that ($ \mathbb{R}_{[x]}$is any polynomial): $L :\mathbb{R}_{[x]}\rightarrow\mathbb{R}_{[x]},(L(p))(x)=p(x)p'(x)$ is not linear transformation. I have already ...
3
votes
1answer
77 views

If all the centralizers are finite, then does it follow that the group is finite?

As far as I have heard, centralizers play an important role in the theory of groups. My question arises from curiosity and the desire to understand how much control centralizers have over the group. ...
3
votes
3answers
138 views

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, ...
0
votes
2answers
100 views

What is an example of a ring $A$ that is isomorphic to ${\Bbb R}^{5}$? [closed]

What is an example of a ring $A$ that is isomorphic to ${\Bbb R}^{5}$ ? Im sorry I am confused about ring theory. Its all new to me.
11
votes
1answer
175 views

Examples of non-isomorphic fields with isomorphic group of units and additive group structure

YACP mentions in a comment that: There are examples of non-isomorphic fields $K$ and $L$ with $(K,+)\cong (L,+)$ and $(K^{\times} ,\cdot)\cong (L^{\times},\cdot)$ Can someone provide an ...
8
votes
2answers
141 views

Counterexample: multiplying modules by elements of an ideal vs. taking linear combinations

Let $R$ be a ring (commutative, unital) and $M$ an $R$-module. Let $I \subset R$ be an ideal. We make the following definitions: $$ A := \{ am \ | \ a \in I,\ m \in M \} $$ $$ B := \left\{ ...
8
votes
2answers
70 views

Maximal ideals in $C^\infty(\mathbb{R})$

I know that for a compact manifold $M$ any maximal ideal in the algebra $C^\infty(M)$ of smooth functions on $M$ is of the form $\mathfrak{m}_p=\{f\in C^\infty(M)|f(p)=0\}$. For example, the proof is ...
9
votes
4answers
670 views

Example of a finite non-commutative ring without a unity

Give an example of a finite, non-commutative ring, which does not have a unity. I can't think of any thing which fits this question. I was thinking $M_2(\mathbb{R})$ but it has the identity. Any ...
1
vote
3answers
57 views

Examples of root, parabolic, and borel subgroups corresponding to roots

I'm interested in seeing a few examples of root, parabolic, and Borel subgroups given a specific reductive group $G$. Here is what I know. Let $G$ be a reductive algebraic group over an ...
6
votes
2answers
115 views

Do silly-rings exist?

A ring can be defined as a near-ring satisfying two-sided distributivity, whose underlying additive group is Abelian. Negating this second stipulation, we obtain the following definition. A silly-ring ...
6
votes
3answers
91 views

Structures with addition, multiplication and exponentiation.

The set $\mathbb{N}$ can be viewed as a mathematical structure with operations off addition, multiplication and exponentiation. Observe that: It forms an Abelian monoid under both addition and ...
10
votes
6answers
342 views

If $G$ is a group and $N$ is a nontrivial normal subgroup, can $G/N \cong G$? [duplicate]

I know $G/N$ is isomorphic to a proper subgroup of $G$ in this case, so the gut instinct I had was 'no'. But there are examples of groups that are isomorphic to proper subgroups, such as the integers ...
3
votes
1answer
35 views

Irreducibility of Lie algebra representations

Let $\mathfrak{g}$ be a finite dimensional real Lie algebra and $\pi: \mathfrak{g} \to \mathfrak{gl}(V)$ be a homomorphism of real Lie algebras where $V$ is a finite dimensional real vector space. But ...
3
votes
4answers
132 views

Number of elements in a group and its subgroups (GS 2013)

Every countable group has only countably many distinct subgroups. The above statement is false. How to show it? One counterexample may be sufficient, but I am blind to find it out. I have ...
2
votes
2answers
54 views

A non-UFD where we have different lengths of irreducible factorizations?

A unique factorization domain (UFD) is defined to be an integral domain where each element other than units and zero has a factorization into irreducibles, and if we have two such factorizations then ...
3
votes
3answers
105 views

Can an object be universal with respect to several properties?

Does anyone know an example of an object $A$ in some category $\mathcal C$ such that $A$ satisfies at the same time more than one universal property? Of course, when I say that $A$ is universal with ...
1
vote
1answer
103 views

Counterexample to Eisenstein criterion

We know Eisenstein criterion about irreducibility of polynomials: if $q(x) = x^n + a_{n-1}x^{n-1} + \dots +a_0 \in \mathbb{Z}[x]$ is such that $\exists p$ prime number with $ p \mid a_{i} \ \forall i ...
4
votes
2answers
173 views

Normal products of groups with maximal nilpotency class

Let $H$ be a nilpotent group of class $a$ and $K$ a nilpotent group of class $b$. If $H$ and $K$ are normal subgroups of a group $G$, then we know that $HK$ is a normal nilpotent subgroup of $G$ and ...
5
votes
1answer
63 views

Counterexample to a lemma about modules

Let R be a ring with identity and not necessarily commutative. Let $M_1, M_2$ be left $R$-modules with submodules $S_1, S_2$ respectively such that $M_1/S_1 \cong M_2$ and $M_2/S_2 \cong M_1.$ Is it ...
5
votes
2answers
436 views

Free modules over commutative rings. [duplicate]

Free modules over a commutative ring $R$ with $1$ have well-defined rank. I have been wondering if there is a ring $R$ such that there are free modules $M'\subset M$ with ...
2
votes
1answer
285 views

Counter examples in group theory

Let $G$ be a finite group with normal subgroups $N_{1}$ and $N_{2}$. Find counter examples to the following statements 1) If $N_{1}\cong N_{2}$ then $G/N_{1}\cong G/N_{2}$ 2) If $G/N_{1}\cong ...
2
votes
1answer
90 views

Objects with a “Homogeneity Principle”

So I don't have to worry about formalities, in the following let $\mathscr{C}$ be a sufficiently nice category--at least nice enough so that the following definition makes sense. I believe concrete ...
9
votes
1answer
249 views

Extending Herstein's Challenging Exercise to Modules

Anybody who has worked through Herstein's Topics in Algebra might remember Exercise 26 of Section 2.5 (in second edition): If $G$ is an abelian group containing subgroups of order $m$ and $n$, ...
2
votes
1answer
69 views

Integral extensions of rings, when one of the rings is a field

The following is from page 61 of Introduction to Commutative Algebra by Atiyah & Macdonald: Proposition 5.7. Let $A\subseteq B$ be integral domains, $B$ is integral over $A$. Then $B$ is a ...
1
vote
2answers
123 views

Must certain rings be isomorphic to $\mathbb{Z}[\sqrt{a}]$ for some $a$

Consider the group $(\mathbb{Z}\times\mathbb{Z},+)$, where $(a,b)+(c,d)=(a+c,b+d)$. Let $\times$ be any binary operation on $\mathbb{Z}\times\mathbb{Z}$ such that ...
1
vote
4answers
130 views

Infinity and structures

Do you know any case (example) where an "infinite" object with a structure (say, an infinite group) cannot be extended (in the sense of adding elements) in any way without it no longer having the ...
2
votes
4answers
153 views

Example of a module for non-mathematicians

I'm looking for a non-trivial1 example of a module that would be recognizable to a non-mathematician. I.e. I'm looking for examples of modules that one may come across in "the real world". The ...
1
vote
3answers
100 views

Ring and Subring with different Identities [duplicate]

Is there an example of a ring $S$ with identity $1_S$ containing a non-trivial subring $R$ which itself has an identity $1_R$, but $1_R\neq 1_S$ (or equivalently $1_S\notin R$). I'd also like to know ...
0
votes
5answers
76 views

Operator with symmetric but without associative?

Addition and multiplication in math, all is symmetric,associative。 But i have no idea about operators with symmetric but without associative. please help me listing any 2-arity operators?
1
vote
2answers
73 views

prove that : $D_{4n} $ is isomorphic to $ D_{2n} \times Z_2 $ when $n$ is odd

let $n$ be odd integer , prove that : $D_{4n} $ is isomorphic to $ D_{2n} \times Z_2 $ it's an example which the text proves ! but i can't understand any thing from the argument ! but i tried to ...
2
votes
1answer
58 views

For every monoid $M$ with zero is there a group $G$ such that $\mathrm{Grp}(G,G)\cong M$?

All monoids that I will consider here have identities. A monoid $M$ is said to have a zero iff $\exists z\in M \forall x\in M (zx=xz=z)$. Let $M$ be any monoid with a zero. Must there exist a group ...
0
votes
1answer
40 views

Sufficient condition for reducibility of polynomial $f(x,y)$

[Dual to this question] Suppose we have a non-constant polynomial $f(x,y)\in\mathbb{Q}[x,y]$ such that the following two conditions are satisfied: 1) For every $x_0\in\mathbb{Q}$, the polynomial ...
2
votes
1answer
86 views

Sufficient condition for irreducibility of polynomial $f(x,y)$

Suppose we have a non-constant polynomial $f(x,y)\in\mathbb{Q}[x,y]$ such that the following two conditions are satisfied: 1) For every $x_0\in\mathbb{Q}$, the polynomial $f(x_0, y)\in\mathbb{Q}[y]$ ...
1
vote
2answers
100 views

Examples of $F$-algebra which has no proper two-sided ideals but has one-sided ideal(s)

Let $F$ be a field and $A$ an $F$-algebra. (And assume that $A$ is finite dimension over $F$ if necessary.) A textbook says that $A$ is simple if it has no proper two-sided ideals. To understand this ...
0
votes
1answer
86 views

Lagrange theorem for finite algebraic structures.

Let $S$ be a finite semigroupoid and let $a\in S$. The minimum of $$\{n\in \Bbb N \mid a^{n+1}=a \}$$ , if it exists, is called the order of $a$ and is denoted by $o(a)$. Which conditions on $S$, ...
138
votes
21answers
12k 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 ...
3
votes
1answer
129 views

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 \begin{equation}A\twoheadrightarrow ...
23
votes
10answers
4k views

Examples of nonabelian groups.

Can anybody provide some examples of finite nonabelian groups which are not symmetric groups or dihedral groups?
10
votes
1answer
145 views

Other Euler characteristics?

At the end of V.3.4 in Algebra: Chapter 0, Aluffi describes the construction of a Grothendieck group over the category of finite dimensional $\operatorname{k}$-vector spaces, ...
4
votes
0answers
62 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
45 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} ...