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9
votes
1answer
342 views

Given a rational number a/b does there exist a finite group G and an automorphism f s.t. f maps exactly a/b elements of G to their own inverses?

I was helping a friend prepare for his intro abstract final and he mentioned the professor had once asked the question name a group and an automorphism that takes 3/4 of the elements of the group to ...
0
votes
1answer
385 views

Weak algebraic structures

The following question can be thought as a sequel of this one. Here I'm looking for a big list of example of weak algebraic structures: here weak means that the structure (i.e. operations) need not ...
2
votes
1answer
519 views

Dedekind Spectra

Is there a class of ring spectra that corresponds to and/or extends the class of Dedekind rings from traditional algebra? Is there a notion of "ring of integers" of a ring spectrum? Additionally, is ...
1
vote
0answers
857 views

Textbook suggestion for advanced algebra? [closed]

After having a solid year long undergraduate course in abstract algebra, I'm interested in learning algebra at a more advanced level, especially in the context of category theory. I've done some ...
5
votes
0answers
274 views

Can any group be realized as the multiplicative group of a ring? [duplicate]

Possible Duplicate: Ring with Z as its group of units? Given a group $G$, does there always exist a ring $R$ such that $R^\times \cong G$? I feel like this isn't true but that's just a hunch. ...
7
votes
2answers
704 views

Virtual algebraic calculation within proofs

It seems to me that the undergraduates I teach have particular difficulty with proofs that involve reasoning about algebraic calculations that arise only theoretically. Since I have in mind doing ...
2
votes
0answers
169 views

Idea behind choosing $\small f(x)$ as $c^{s}x^{p-1} \frac{[\theta(x)]^{p}}{(p-1)!}$ in the proof that $\pi$ is transcendental.

I am going through the article at this link, where the author proves that: "$\pi$ is $\text{transcendental}$ over $\mathbb{Q}$". Although, I understand the proof, I have some doubts. At page $6$, ...
3
votes
2answers
377 views

Double orthogonal complement of a finite module

Crossposted from math.stackexchange since I'm not getting any answer. Let $W$ be the finite $\mathbb{Z}$-module obtained from $\mathbb{Z}_q^n$ with addition componentwise where $\mathbb{Z}_q$ is the ...
9
votes
2answers
888 views

Example of a Group which has $\text{SL}_{n}(\mathbb{Z})$ as the automorphism group

For the past one week, I have been trying to learn more about Automorphism groups of different groups. Very recently one of my friend asked this question to me: What is the Autmorphism group of ...
1
vote
1answer
246 views

Defect groups and subgroups

I have asked this question on Stack Exchange but had no response; it's been bugging me for a few days. I am struggling to see how to apply Mackey's theorem to prove a certain Lemma in Local ...
5
votes
2answers
1k views

Example of an infinite abelian but non-cyclic group whose automorphism group is cyclic.

Can anyone give me an example of: An infinite abelian but non-cyclic group whose automorphism group is cyclic.
10
votes
1answer
313 views

Formally undecidable problems on finitely presented quandles

In the literature, one sometimes sees the claim that finitely presented quandles (in particular, knot quandles) are "hard to deal with". Hence, a great deal of effort has gone into studying finite ...
4
votes
2answers
4k views

An image of the hierarchy of algebraic structures

Hello! Does anybody know an image of a graph featuring the hierarchy of algebraic structures? Something rather complete. So far I've found similar images describing the hierarchies of ...
2
votes
3answers
545 views

Are these Two Definitions of Quadratic Form (Algebraic, Topological) Related to Each Other?

Hi, All: I am trying to see if there is a nice relation between two different definitions of quadratic form q; a topological definition $q_T$, and an algebraic definition $q_A$, and, if there is, how ...
1
vote
2answers
393 views

Difference between orthogonal form and seminormal form

Frequently in the literature on Hecke algebras for the symmetric group and their generalisations, one encounters references to Young's seminormal form and Young's orthogonal form. I have a good ...
0
votes
3answers
2k views

Suggestions for a good abstract algebra book [closed]

i am early undergraduate looking for a good textbook gor algebra, i don't want a too wordy book@@
7
votes
2answers
2k views

fgf = f, gfg = g, fg not necessarily identity, what was that called?

A very simple question, I just totally forgot how it was called, and google is not helping. There's a pair of functions $f:X\to Y$, $g:Y\to X$. $fgf = f$, $gfg = g$, but $fg$ and $gf$ don't need to ...
0
votes
2answers
292 views

Aut(G) acting transitively on a finite group [closed]

How Aut(G) acting transitively on a finite group G^* can lead G to be elementary abelian group? G^*=G-{1}.
0
votes
1answer
401 views

Number of ideals in primary decomposition

Dear friends, I have the following question: Let K be an algebraic number field and [K:Q]=n. Let O_K be a full ring of integers of K. Assume that O \subset O_K is a subring such that rank of O over Z ...
0
votes
1answer
548 views

Groups GLn(F) and PSLn(F)

As J.S.Rose noted in his book "A Course on Group Theory" : There is a section of GLn(F) which is isomorphic to PSLn(F), n≥1, F is a field"?. I ask that "What can this section be?"
3
votes
3answers
547 views

About solvable groups

Is it possible for a group (non-simple and non-abelian) that solvability of all of its proper subgroups leads the whole group to be solvable?
0
votes
2answers
210 views

The X-series (for groups)

It goes without saying that the name in the title tentatively refers to a series whose name one does not know yet and probably in the future I may come with a post titled "The x-sequence" or "The ...
1
vote
1answer
340 views

Unimodular column property

Hi, I know that if $R$ is a ring such that every projective $R$-module finitely generated is free then $R$ has the unimodular column property. I would like to know if there is a ring $R$ that doesn't ...
7
votes
1answer
660 views

Direct sum of injective modules over non-Noetherian rings

Hi. I know, by the Bass-Papp theorem, that if every direct sum of injective $R$-modules is injective then $R$ is Noetherian. I would like to know if there exists a direct sum of injective $R$-modules ...
5
votes
5answers
2k views

motivation of filtered colimits

I am trying to move in categorical algebra beyond the basics. A Lawvere theory L is a small category with finite products. (I know that there also is a functor $(skeleton(FinSet))^{op}\to L$, which ...
2
votes
2answers
349 views

Related to fractional ideals

$K$ a field, $A\subset K$ a subring, $M\subset K$ an $A$-submodule. Define $$(A:_{K}M):= \lbrace s\in K|sM\subset A\rbrace$$ Then it is easy to see that $$M\subset A\Longleftrightarrow A\subset ...
6
votes
3answers
835 views

Does “finitely presented” mean “always finitely presented”, considered in general

I'm wondering about the question, "If we have a finitely presented __, is it necessarily finitely presented with respect to any finite generating set for it?" I know this is true for groups and for ...
12
votes
4answers
2k views

Unique factorization domains

Let $c$ be an integer, not necessarily positive and $|c|$ not a square. Let $\mathbb{Z}[\sqrt{c}]$ be the set of complex numbers $$a+b\sqrt{c}, a, b\in \mathbb{Z},$$ which form a subring of the ring ...
0
votes
1answer
136 views

Projectively splitting module

Is there a name for such class of modules $M$ such that $M\rightarrow N\rightarrow 0$ splits for every $N$ ?
-1
votes
2answers
544 views

Unit ideal in non-commutative rings [closed]

In a non-commutative ring (with identity), is it possible for an element which does not possess left or right inverses to generate the entire ring? i.e. $(r)=R$, where (r) is the two-sided ideal ...
0
votes
3answers
1k views

Union of two proper subgroups [closed]

When can a group be written as the set-theoretic union of its proper subgroups?
3
votes
1answer
1k views

Bounds for zeros of polynomials with only real zeros

The formula in http://de.wikipedia.org/wiki/Diskussion:Nullstelle#References gives upper and lower bounds x1 and x2 for the roots of a polynomial all of whose roots are real. Where can I find a ...
14
votes
2answers
2k views

Does any textbook take this approach to the isomorphism theorems?

Below, I present an outline of a proof of the first isomorphism theorem for groups. This is how I usually think of the first isomorphism theorem for ______, but groups will get the points across. My ...
5
votes
1answer
323 views

What is the Schouten bracket for the Chevalley-Eilenberg complex with coefficients in a nontrivial module?

Let $\mathfrak g$ be a Lie algebra. The Chevalley-Eilenberg complex is defined to be $\wedge^* \mathfrak g$ with differential $d\colon \wedge^* \mathfrak g\to \wedge^{*-1}\mathfrak g$ defined by ...
8
votes
4answers
631 views

Groups and rings which are not sets

An algebraic structure such as a group, ring, field, etc. is usually defined to be a set with some operations satisfying certain properties. I am curious what, if anything, goes wrong when the ...
14
votes
1answer
761 views

Is Dependent Choice equivalent to the statement that every PID is factorial?

In this question, it was asked if AC is needed in the proof of the well-known fact that every principal ideal domain is factorial. As KConrad and Joel David Hamkins have pointed out, only DC, the ...
71
votes
18answers
6k views

How do you decide whether a question in abstract algebra is worth studying?

Dear MO-community, I am not sure how mature my view on this is and I might say some things that are controversial. I welcome contradicting views. In any case, I find it important to clarify this in my ...
2
votes
7answers
2k views

How to tell if two random polynomials are identical

Let t be a positive real number. Let P(x) and Q(x) be two random polynomials with integer coefficients. If P(t) = Q(t), then what is the probability that P(x) is not identical to Q(x)? Will it make a ...
4
votes
1answer
1k views

Non-trivial consequences of Baer's theorem and Lucchini's theorem in subnormality theory

There are a couple of beautiful results in finite group theory that look trivial, at least on a first glance, but require non-trivial facts to prove. I am basically interested in whether these results ...
2
votes
2answers
250 views

Admissible finite groups

I want to know when an abelian group of even order is admissible (or has a complete map)? And when a nonabelian group of even order is admissible (or has a complete map)? Thanks.
2
votes
1answer
513 views

Rank of a linear combination of quadratic forms

Suppose we have a set of quadratic forms $Q_i (x_1, \dots, x_n)$ for $1 \leq i \leq k$ in $n$ variables, defined over $\mathbb{R}$. We suppose these are 'collectively nondegenerate' in the sense that ...
17
votes
6answers
2k views

Consequences of not requiring ring homomorphisms to be unital?

As defined in many modern algebra books, a homomorphism of unital rings must preserve the unit elements: $f(1_R)=1_S$. But there has been a minority who do not require this, one prominent example ...
3
votes
5answers
539 views

“Convex Optimization” over varying-dimension vector space?

For all instances of Convex Optimization I know of, the dimension of the vector space is defined beforehand. Is there an area of mathematics that deals with "convex optimization" of varying-dimension ...
4
votes
2answers
2k views

Sylow's theorem 3rd Proof Page 96 I.N.Herstein

I was just going through the 3rd Proof of Sylow's theorem given in the "Topics In Algebra" Book by I.N. Herstein. It looked very interesting and i really liked its Philosophy. My question what is its ...
4
votes
5answers
2k views

Indeterminate “x” in Abstract Algebra/Ring Theory [closed]

How do you interpret the indeterminate "x" in ring theory from the set theory viewpoint? How do you write down R[x] as a set? Is it appropriate/correct to just say that $R[x] = \{ f: R \to R | ...
1
vote
2answers
837 views

Extension problem

As I understand, if $0\rightarrow A\rightarrow X\rightarrow B\rightarrow 0$ is a short exact sequence of abelian groups, $\mbox{Ext }_{\mathbb{Z}}^{1}(B,A)$ gives all the isomorphism classes of what ...
10
votes
4answers
1k views

Linearly independent subsets of a free module

Is it true that the cardinality of every maximal linearly independent subset of a finitely generated free module $A^{n}$ is equal to $n$ (not just at most $n$, but in fact $n$)? Here $A$ is a nonzero ...
21
votes
6answers
2k views

How fast are a ruler and compass?

This may be more of a recreational mathematics question than a research question, but I have wondered about it for a while. I hope it is not inappropriate for MO. Consider the standard assumptions ...
1
vote
1answer
917 views

Cardinality of symmetric group [duplicate]

Possible Duplicate: Cardinality of the permutations of an infinite set Why does the symmetric group on an infinite set X have the cardinality of the power set ${\cal P}(X)$?
25
votes
3answers
2k views

Feit-Thompson Theorem: The Odd Order Paper

For reference, the Feit-Thompson Theorem states that every finite group of odd order is necessarily solvable. Equivalently, the theorem states that there exist no non-abelian finite simple groups of ...