Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, modules, among other topics.

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Proving reflexivity from transivity and symmetry.

Property 2 of an equivalence relation states that if $a\sim b$ and $b\sim c$ then $a\sim c$. What is wrong with the following proof that properties 2(symmetry) & 3 (transitivity) imply ...
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Homomorphism of Free Groups

I am reading the theorem of homomorphism of free group from Fraleigh's text in $\S$36 and could only get a fuzzy idea at best: Let $G$ be generated by $A = \{a_i \mid i \in I \}$ and let $G'$ be ...
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What is an example that Euclidean function varies by a unit?

Let $R$ be a Euclidean domain and $f$ be a Euclidean function on $R$. (Not necessarily submultiplicative) Let $a,b$ be nonzero elements of $R$ such that $a=ub$ where $u$ is a unit of $R$. Is there ...
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How do we define how many lines lie on a given hypersurface in $\mathbb{F}_q^n$

Given the following surface, for example: $$\mathbb{H} = \{(z_1,z_2,z_3,z_4,z_5)\in \mathbb{F}_{p^2}: z_1 - z_1^p - z_2z_3^p + z_2^pz_3 - z_4z_5^p + z_4^pz_5 = 0\}$$ in $\mathbb{F}^5_{p^2}$. We know ...
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What is the significance of $A+ (B\cap C)=(A + B)\cap C$, where $A\subseteq C$, for modules?

My book (Introduction to Ring Theory, Paul Cohn) states this as a theorem and gives a proof. The book usually skips over trivial/easy proofs, so I don't really understand why this is in here. Isn't ...
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Algebraic structures and axiomatic systems

In one textbook appears the following sentence: An algebraic structure is a nonempty set $M$ together with one or more operations (i.e. a function $*:M\times M\rightarrow M$) which satisfy some ...
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Let $F$ be a finite field. Determine all $x\in F$ s.t .: $x^2=1$

What I've done so far: This means that $x$ is auto-inverse with respect to multiplication. I've seen that this holds for $x=1$, since $1\cdot 1=1$, and $x=(-1)$, since $(-1)(-1)=1$ (according to ...
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Does there exist such an invertible matrix?

Let $n \geq 1$ and $A = \mathbb{k}[x]$, where $\mathbb{k}$ is a field. Let $a_1, \dots, a_n \in A$ be such that $$Aa_1 + \dots + Aa_n = A.$$ Does there exist an invertible matrix $\|r_{ij}\| \in ...
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23 views

Orbits for a subgroup $H$ of $G$ acting on $G$

Why the orbits for a subgroup of $H$ of $G$ acting on $G$ by left multiplication are the right cosets of $H$ in $G$?
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60 views

Show that $x^3-3$ is irreducible in $\Bbb Z_7[x]$.

Show that $x^3-3$ is irreducible in $\Bbb Z_7[x]$. In the text, we haven't gotten to the theorem that the roots of polynomials are the only factors , and I would rather not prove it in this ...
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Let $a,b \in$ group $G$ such that $ab=ba, \gcd(O(a),O(b))=1$. Prove that $O(ab)=O(a)O(b)$.

My attempt: Let $O(a)=m, O(b)=n$, then $mx+ny=1$ Let $O(ab)=p$, then using commutative property, $(ab)^p=a^pb^p=e$, which is the identity. Then $a^p=e, b^p=e$, hence, $m | p$ and $n | p$. So, ...
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proving Caley's Theorm using the conjugacy operation instead of left multiplication

I need to prove Caley's theorem that each group G is isomorphic to a sub group of S(n). Wherever I check it is proven using the operation of multiplying from left side. that means $f_g(x) = g*x$ I ...
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show that $\varphi : H \to H /N$ is one-to-one correspondence using kernel

I have a group G and subgroup N that is normal to G. As a part of proving the 3rd isomorphism theorm (a version of it) I need to prove that the transformation from the subgroups of G containing N ...
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Under what conditions is the homology of a dg coalgebra a graded coalgebra?

I'm trying to get a feel for some differential graded (dg) structures. Suppose $C$ is a differential graded coalgebra over a commutative ring $k$, i.e. a graded $k$-module equipped with a coproduct ...
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40 views

A question about a proof of $\operatorname{ord}(\mathbb{Q}/\mathbb{Z})<\infty$

Let $r+\mathbb{Z}$ be some coset of $\mathbb{Q}/\mathbb{Z}$, where $r\in\mathbb{Q}$, thus we can write $r$ as ratio $\frac{m}{n}$, where $n$ is a positive integer. Then $n$ times $r+\mathbb{Z}$ is ...
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Uniqueness in existence of a bilinear form

Let $E$ be a finite dimensional vector space over field $\mathbb R$ with $E^*$ as the dual space of $E$.($\dim E=n$) $\Omega^P(E):=\{ \alpha\colon \overbrace{E\times\cdots\times E}^{p- ...
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32 views

Quotient group of free groups

Let $G=\langle g_1,\ldots,g_k\rangle$ be a free abelian group generated with $g_1,\ldots,g_k$ and let $H=\langle g_{r+1},\ldots,g_k\rangle$ be a free abelian subgroup of $G$. Is it then the case that ...
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Confusion with Centers, Conjugacy Classes, and Normal Subgroups

Ok this is a bit of a soft question, but I am in my first semester of algebra and getting continually confused by (what seems to me) some competing sets. Let $G$ be a group The center of $G$ is ...
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Degree of minimal polynomial for $\sin (\frac {2 \pi} 7)$

So I was playing around with trying to prove the regular 7-gon is not constructable under qualifier-exam conditions, so I didn't have a book open. I got it down to having (If I didn't make any basic ...
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Left Coset of $H = \lbrace \epsilon, \alpha \rbrace$

"A Book of Abstract Algebra" presents the following exercise: In each of the following, $H$ is a subgroup of $G$. For each coset, list the elements of the coset. $$G=S_3, H= \lbrace \epsilon, ...
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Relations between the Latin squares of order n and the groups of order n.

Given a group $g$ of order $n\in \mathbb{N}$, it is clear that its Cayley table has the Latin square property. That is, every row and column of the Cayley table contains precisely the elements of $g$. ...
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How many points does the surface $\mathbb{H}$ defined with the stated expression contain in $\mathbb{F}^5_{p^2}$?

How many points does the surface $$\mathbb{H} = \{(z_1,z_2,z_3,z_4,z_5)\in \mathbb{F}_{p^2}: z_1 - z_1^p - z_2z_3^p + z_2^pz_3 - z_4z_5^p + z_4^pz_5 = 0\}$$ contain in $\mathbb{F}^5_{p^2}$? ...
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50 views

Curve of genus $g$ with a point removed

Let $C$ be a smooth projective curve of genus $g$. If we pick two distinct points $p,q\in C$, when are $C\setminus\{p\}$ and $C\setminus\{q\}$ isomorphic or not isomorphic? When $g = 0$, they are ...
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How many $2$-Sylow subgroups in $G$ with $|G| = 2^2\cdot 3$?

I have a group $G$ with $|G| = 2^2\cdot 3$. I also know it has $4$ Sylow-$3$ subgroups. I need to show that there is $1$ Sylow-$2$ subgroup. (This is all I have left from the full question.) Any ...
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106 views

Every group of order $150$ has a normal subgroup of order $25$

Let $G$ be a group of order $150$. I must show that it has a normal subgroup of order $25$. The hint says to show that is has a normal subgroup of order $5$ or $25$. Now from Sylow, I know that the ...
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If $L$ splits $D=I\otimes N$ does $L$ split $I$ and $N$?

Let $D$ be a central simple division algebra over the field $F$. Let $D\sim I\otimes N$ where $I$ and $N$ are division algebras in Br$(F)$ and $\sim$ is equivalence in Br$(F)$. I am interested in this ...
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$O_S$ is the integral closure of $k[T]$ in $F$ for some embedding of $k(T)$ in $F$?

Let $F$ be a function field in one variable over a field $k$. Let $S$ be a nonempty finite subset of the set of all places of $F$. Let$$O_S = \{f \in F: \text{ord}_v(f) \ge 0 \text{ for all }X ...
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Why is it true that $|AB:A|=|B:A\cap B|$ even if $A$ is not normal in $AB$? (Second Isomorphism Theorem)

I just read about the First and Second Isomorphism Theorems in the book Abstract Algebra by Dummit and Foote. After proving the Second Isomorphism Theorem, they said: Proposition 13 isn't ...
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Well-defined map from $G/N$ to $G$ that is a homomorphsim?

Let $G$ be a group, and let $N$ be a normal subgroup of $G$. I know that the natural projection homomorphism is a surjective homomorphism from $G$ onto $G/N$. If I choose a particular representative, ...
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If a set $S$ generates an ideal $I\subset F[x_1,x_2,\ldots,x_n]$, then there is a finite subset $S_0 \subseteq S$ which generates $I$

The question: If $I$ is an ideal in $F[x_1,x_2,\ldots,x_n]$ generated by a set of polynomials $S$, then there is a finite subset $S_0 \subseteq S$ which generates $I$. By the Hilbert Basis ...
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When the trace of the Frobenius homomorphism is zero?

Let's consider an elliptic curve over a finite field $\mathbb F_p$. The trace of the Frobenius homomorphism is defined as: $$a_p=p+1-\#E(\mathbb F_p)$$ See for example here. I read that this value ...
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Normality of a normal subgroup of normal subgroup of G

Let $G$ be a non-Abelian Group and $H$ is normal subgroup of $G$. Is it always true that a normal subgroup $K$ of $H$ is also normal in $G$? Justify your answer. My answer is that, this is not true ...
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What Are the Indeterminates in the Polynomials in This Proof?

To prove that every field $F$ has an algebraic closure, the first step is to prove that there exists an extension $F_1$ such that each polynomial in $F[x]$ has a root in $F_1$. Whereas in the ...
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What is the structure of $S$?

Suppose we define an equivalence relation on $\mathbb R$ by $aRb$ iff $\{a\}=\{b\}$ for $a,b\in\mathbb R$. Here $\{.\}$ defines the fractional part. In other words, $aRb$ iff $a-b\in\mathbb Z$. ...
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Is ${(k^n)}^{\otimes r}$ a faithful $k\Sigma_r$-module for $n\geq r$?

I have the following question: Let $k$ be an infinite field. Let $E:={(k^n)}^{\otimes r}$ and let the symmetric group $\Sigma_r$ act from the right on $E$ by place permutations. It is well-known that ...
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Is this parabolic induction?

In one of my previous questions, @PL. explained the idea of parabolic induction. In my bachelor thesis I used a technique quite like this, to find all irreducible representations of a certain group. I ...
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Given $K(\alpha)/K$ and $K(\beta)/K$ abelian extensions, prove that $K(\alpha + \beta)/K$ is an abelian extension.

Problem: Let $K(\alpha)/K$ and $K(\beta)/K$ algebraic field extensions so that their respective Galois groups are abelian. Prove that the Galois group of the field extension $K(\alpha + ...
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Show that $R = \bigcap_mR_m$ whenever $R$ is an integral domain

Show that $R = \bigcap_mR_m$ whenever $R$ is an integral domain, where the intersection is indexed by all maximal ideals of $R$. $R \subset \bigcap_mR_m$ is clear since $R \subset R_m$ for all $m$ ...
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Question concerning a property of polynomial functions on $\Gamma:=\text{GL}_n(K)$ and the Schur algebra

I'm reading Green's book ''Polynomial Representations of GL_n. with an Appendix on Schensted Correspondence and Littelmann Paths''. Consider theorem 2.4b) part (i) on page 14: Consider the map $e : ...
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How to prove that the Schur algebra is isomorphic to a certain endomorphism ring?

I'm reading Green's book ''Polynomial Representations of GL_n. with an Appendix on Schensted Correspondence and Littelmann Paths''. Consider theorem 2.6c) on page 18: Let $K$ be an infinite field ...
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Question concerning a self-injective algebra and a faithful module

I'd like to know how corollary 2.11 of http://www.sciencedirect.com/science/article/pii/S002186930098726X# follows from theorem 2.10 from the same reference. 1) I know that $A$ being self-injective ...
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I have to show $A$ is not cyclic

Suppose that $A=C\oplus C=\begin{pmatrix} C & 0 \\ 0& C \end{pmatrix}$, $C$ be a companion matrix of $m(x)=m_0+m_1x+\ldots+m_{n-1}x^{n-1}+x^n$ . I have to show that $A$ is not cyclic. Can any ...
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Prove that $\mathbb Z^n$ is not isomorphic to $\mathbb Z^m$ for $m\neq n$

Prove that groups $\mathbb Z^n$ is not isomorphic to $\mathbb Z^m$ for $m\neq n$ My try: Let $\mathbb Z^n\cong \mathbb Z^m $ .To show that $m=n$. Case 1:Let $m>n$.Now that $\mathbb Z^m$ has $m$ ...
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Prove a cycle of length l is odd if l is even? [closed]

This is my first course on Group Theory. How do I go about proving this?
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Showing Sylow $p$-groups are Abelian if $n_p=2p+1$

An old qual question on Sylow $p$-groups: Assume that $p$ is an odd prime and $G$ is a finite simple group with exactly $2p+1$ Sylow $p$-groups. Prove that the Sylow $p$-groups of $G$ are abelian. ...
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21 views

What is the purpose of the almost maximal and $ p $-supersoluble subgroup?

Suppose that $ H \unlhd G $ such that $ G/H $ is supersoluble. Suppose that there is an element $ y \in H $ such that $ H = \langle y \rangle L $ for any almost maximal subgroup L of $ H $; then $ G $ ...
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1answer
28 views

Given $A$-modules $N \subset M$ such that $N_m=M_m$ for all maximal ideals $m$, show that $M=N$

I am working on this exam question 6 $A$ is commutative ring with $1$ a) If $N \subset M$ are $A$-modules and $N_m=M_m$ for all maximal ideals $m$, show that $M=N$. We know that $N_m=M_m$ ...
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24 views

If $A=\bigcup F_i$ is a filtered algebra, why is the multiplication on $\mathcal{G}(A)=\bigoplus (F_n/F_{n-1})$ well-defined?

From the wiki page, suppose $A=\bigcup F_i$ is a filtered algebra, and $\mathcal{G}(A)=\bigoplus_{n\geq 0}G_n$ the associated graded algebra, where $G_0=F_0$ and $G_n=F_n/F_{n-1}$. The ...
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59 views

how do I prove that $S_4$ has no normal subgroup of order 6

Let $N$ be a normal subgroup of $S_4$. I have proven that $|N|\ne 2,3,8$. Yet, I don't know how to prove that $|N|\neq 6$. Should I compute all subgroups and check this?
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List Elements of Left/Right Cosets of $H$ for: $G=\mathbb{Z}_{15}, H=\langle 5 \rangle $

"A Book of Abstract Algebra" presents this exercise: In each of the following, $H$ is a subgroup of $G$. List the cosets of $H$. For each coset, list the elements of the coset. ...