# Questions tagged [abstract-algebra]

For questions about groups, rings, fields, vector spaces, modules and other algebraic objects. Associate with related tags like group-theory, ring-theory, modules, etc. to clarify which topic of abstract algebra is most related to your question and help other users when searching.

57,551 questions
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### Isomorphy of quotient modules implies isomorphy of submodules

Let A be some commutative ring, $M$ an $A$-module and $N_1, N_2$ two submodules of $M$. If we have $M/N_1 \cong M/N_2$, does this imply $N_1 \cong N_2$? This seems so trivial, but I just don't see a ...
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### Why $\gcd(qb+r,b)=\gcd(b,r)$?

Given: $a = qb + r$. Then it holds that $\gcd(a,b)=\gcd(b,r)$. That doesn't sound logical to me. Why is this so? Addendum by LePressentiment on 11/29/2013: (in the interest of http://meta.math....
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### Dedekind domain with a finite number of prime ideals is principal

I am reading a proof of this result that uses the Chinese Remainder Theorem on (the finite number of) prime ideals $P_i$. In order to apply CRT we should assume that the prime ideals are coprime, i.e. ...
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### $\mathbb{Q}$ is an injective $\mathbb{Z}$-module

I just learned what an injective module is and I want to consider some basic examples. Apparently, $\mathbb{Q}$ is an injective module over $\mathbb{Z}$, but I can't find an elementary proof of this ...
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### First Isomorphism Theorem in Generality [duplicate]

Possible Duplicate: Quotient objects, their universal property and the isomorphism theorems In how general of a setting is the first isomorphism theorem true? Wikipedia states it in terms of ...
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### concrete isomorphisms of polynomial rings

Question 1: In A Singular Introduction to Commutative Algebra, page 222, there is written: How can I check that this isomorphism actually holds? I would really prefer a computational proof (using a ...
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### Are free products of finite groups virtually free?

Is the free product $A*B$ of two nontrivial finite groups always virtually free? If yes, is it easy to show?
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### Commutative rings without assuming identity

I was going through Exercises in Dummit&Foote, which does not assume identity in the definition of a ring, and reached the following exercise: Prove that in a Boolean ring ($a^2 = a$ for all $a$...
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### Determine a generator $g$ of the group $\mathbb{Z}_p^\times$ given some elements and their inverses

Is there a way of finding a generator of a multiplicative group $G = \mathbb{Z}_{41723027}^\times$ given some elements of the group : S = $\{ 4, y=1063, 1064, y^{-1}=12049830, 41723026 \}$ In the ...
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### Why do we assume $a+b=b+a$ in a Ring with 1. Also, is it true with Rings without 1?

I was wondering why do some people use redundant axioms in definitions? If you just expand $(a+1)(b+1)=(a+1)b+a+1=ab+b+a+1$ $(a+1)(b+1)=a(b+1)+b+1=ab+a+b+1$. Hence, $ab+a+b+1=ab+b+a+1$, then cancel ...
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### Proof that no permutation can be expressed both as the product of an even number of transpositions and as a product of an odd number of transpositions

I am aware that there are a couple of well-known proofs of this theorem, but I'm specifically grappling with the proof given in Fraleigh's A First Course in Abstract Algebra (Theorem 9.15 in the ...
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### Symmetric group [duplicate]

Possible Duplicate: What kind of “symmetry” is the symmetric group about? Could you tell me please, why Symmetric group is called "symmetric"? I found an example with quadrate, where ...
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### Existence of subgroup of order six in $A_4$

Show that the alternating group $A_4$ of all even permutations of $S_4$ does not contain a subgroup of order $6$. For me am thinking to write all elements of $A_4$ and trying to find every ...
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### How to find kernel of this map: is my attempt correct?

Find $\ker(\varphi)$ and $\varphi(3,10)$ for $\varphi \colon \mathbb Z\times\mathbb Z\to S_{10}$ such that $\varphi(1,0)=(3,5)(2,4)$ and $\varphi(0,1)=(1,7)(6,10,8,9)$. For example if I get order ...
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### Somebody help me how to find $\ker(Q)$ and $Q(21)$

In my study of group homomorphism I find a question like this. Let $Q\colon \mathbb Z\to S_8$ be group homomorphism such that $Q(1)=(1,4,2,6)(2,5,7)$. Then find $\ker(Q)$ and $Q(21)$. In my mind I ...
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### Proof of Yoneda Lemma [closed]

Can anyone explain to me Yoneda Lemma proof in great details? i.e. they usually say " ... it is easy to see that these morphisms are inverse to each other.." without explanation.
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### How to find the elements of $S_n$?

Consider $S_n$, I do not understand how to get the elements of $S_n$ especially for $n \geq 4$ Any one to show me how simply I can find them. Thanks a lot
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### If $R$ is an integral domain, and every $R$-module is projective, must $R$ be a field?

Let $R$ be an integral domain with the property that all modules over $R$ are projective. Does it follow that $R$ is a field? Obviously the converse is true.
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### Intuitive understanding regarding the number of roots of a polynomial over a field and invertibility

In considering: A polynomial over a field of degree n has at most n roots. -- How does this make use of the stipulation "over a field" - especially with an eye toward inveritbility ? Is it to insure ...
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### If an Abelian group $G$ has order $n$ and at most one subgroup of order $d$ for all $d$ dividing $n$ then $G$ is cyclic

If an Abelian group $G$ has order $n$ and at most one subgroup of order $d$ for all $d$ dividing $n$ then $G$ is cyclic. I am trying to use the structure theorem for finitely generated abelian groups....
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### Normal sylow subgroup of a finite group is characteristic

Let $P$ be a normal sylow $p$-subgroup of a finite group $G$. Since $P$ is normal it is the unique sylow $p$-subgroup. I would like to say if $\phi$ is an automorphism then $\phi(P)$ is also a sylow ...
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