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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A question on coalgebras(1)

Is there a complex coalgebra $C$ with dimension at least 2 for which the scalar operators $T(x)=\lambda x$ are the only operators which satisfy $$(T\otimes T)\circ \Delta= \Delta \circ T^{2}$$ ...
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How do you evaluate elements of the Drinfeld double $D(H)$ against elements of $H^\ast$ or $H$?

Proposition 8.2 of Majid's primer on quantum groups says that if $H$ is a finite dimensional Hopf algebra with quantum double $D(H)$, then this is a factorizable Hopf algebra with quasi-triangular ...
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1answer
82 views

Why this $\sigma \pi \sigma^{-1}$ keeps apearing in my group theory book? (cycle decomposition)

I'm studying cycle decomposition in group theory. The exercises on my book keep saying things like: Find a permutation such that $\sigma (1 2) \sigma^{-1} = (123)$ Prove that there is no permutation ...
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1answer
34 views

Of the 16 binary operations on a two element set, which ones are commutative, associative, have an identity element, and have inverse?

If you you 16 binary operations $$(a*a)=a$$ $$(a*b)=a$$ $$(b*a)=a$$ $$(b*b)=a$$ $$(a*a)=a$$ $$(a*b)=b$$ $$(b*a)=a$$ $$(b*b)=b$$ $$(a*a)=b$$ $$(a*b)=a$$ $$(b*a)=b$$ $$(b*b)=a$$ $$(a*a)=b$$ ...
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Example for 2 Modules(M,N) where the set of the m$\otimes$n is not a submodule

If I remember my Algebra class correctly, this is in general not true because, for example, the addition isn't closed. So if I take: $B:\mathbb{R}^2 \times \mathbb{R}^2\to \mathbb{R}^{2^2}\cong ...
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1answer
31 views

Prove that there is no permutation $\gamma$ such that $\gamma (1 2) \gamma^{-1} = (1 2 3)$

I need to prove that there is no $\gamma$ such that: $$\gamma (1 2) \gamma^{-1} = (1 2 3)$$ First of all, I'll try to write $\gamma$ in a generic way: $$\gamma = (a b c) \implies \gamma^{-1} = ...
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1answer
24 views

Sizes of Quotient Rings of DVRs with Finite Residue Field

If $R$ is a discrete valuation ring (DVR) with maximal ideal $\mathfrak{m}$ such that $R/\mathfrak{m}$ is finite, then all quotient rings of $R,$ namely $R/\mathfrak{m}^n$ for $n \in \mathbb{N},$ are ...
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Example of $S[1/a] \cong S[1/b]$ as rings via $\phi$, where $S$ is a UFD, $a, b \in S$, and $\phi(U(S)) \neq U(S)$.

Above, $U(S)$ refers to the units of $S$. This problem stems from reading a paper titled "Translates of Polynomials" from 2000, where a fact about a ring isomorphism between $S[1/a]$ and $S[1/b]$ is ...
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26 views

Can two representations with different dimensions be isomorphic?

For a finite group G and two irreducible representations, with different dimensions. How would I show that they can not be isomorphic?
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Generalization of Singular locus and non-free locus to an algebra

Let $R$ be a commutative noetherian local ring with maximal ideal $\mathfrak{m}$ and $ \Lambda $ be a noetherian $ R $-algebra. Recall that: (1) The singular locus of $R$, denoted by $\mathsf{Sing} ...
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problem in meaning of symbol in commutator subgroup

i was reading paper "OUTER AUTOMORPHISMS IN NILPOTENT p-GROUPS OF CLASS 2, H. LlEBECK" in page 2 there is a symbol i dont get. if G is generated by a basis $a_λ, λ∈Λ$ and z∈Z⋂Φ(G) for σ(z,μ) be ...
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4answers
53 views

If $G$ is a field and there is an isomorphism $f\colon H/I \to G$, then does $I$ have to be a principal ideal?

I noticed that the ideal $I = \left(2, 1 + \sqrt{-7}\right)$ follows the definition of a non-principal ideal. I took two random elements from $\mathbb Z[-7]$, say $ a + b\sqrt{-7}$ and $ c + ...
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26 views

Indecomposable commutative rings [on hold]

Let $R$ be a commutative ring. Can we say that $R=\bigoplus_{i\in I}R_i$ or $R=\prod_{i\in I}R_i$ where $R_i$ are commutative ring and $I$ is an infinite set?
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1answer
22 views

A simple divisible module

Let $k$ be division ring and $V$ be a left $k$-vector space of infinite dimension. Let $E=\text{End}_k(V)$ be the ring of right linear transformations on $V$. My questions are: (1) Is $V$ a simple ...
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1answer
64 views

Are there any free or fascist boolean algebras?

A boolean algebra is an algebra with the binary operations $\wedge$, $\vee$, an unary operation $\neg$, and constants $0$, and $1$, satisfying axioms. A heyting algebra is an algebra with the binary ...
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553 views

Why do we have “another” definition for the kernel?

Why does the definition $\ker(f)=\{(a,a')\in A\times A: f(a)=f(a')\}$ exist? This definition is for any sort of algebraic system and any sort of function. But which came first... this definition or ...
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32 views

Why is the coproduct $\Delta$ of the quantum double $D(H)$ an algebra homomorphism?

If $H$ is a a finite dimensional Hopf algebra, $H^\ast\otimes H$ has a Hopf algebra structure with multiplication $$ (\phi\otimes h)(\psi\otimes g)=\sum \psi_2\phi\otimes h_2g\langle ...
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1answer
60 views

Frobenius automorphism and non-split Cartan subgroup of $\mathrm{GL}_2(\mathbb{F}_p)$

For completeness I give some definitions. Let $p$ a prime number and consider $V$ a 2-dimensional $\mathbb{F}_p$-vector space. Consider $k$ a sub-algebra of $\mathrm{End}(V)$ that is a field of $p^2$ ...
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59 views

Failure of group definition with weaker axioms

In "A First Course in Abstract Algebra", Edition 7, p.43, Fraleigh writes that It is possible to give axioms for a group $\left<G,*\right>$ that seem at first glance to be weaker, namely: ...
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1answer
42 views

Show that $(t^m-1)/(t^n-1)$ is a square if and only if $(\exists s \in \mathbb{Z})\ m=np^s$

I want to show the following lemma: Assume that the characteristic of the field $F$ is $p$ and $p>2$. Then $(t^m-1)/(t^n-1)$ is a square in $F[t, t^{-1}]$ ($F[t,t^{-1}]$: the polynomials in ...
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40 views

Polynomial Interpolation and Data Integrity

This question is about polynomial interpolation and security. Please consider a scenario where we have a polynomial $f$, one of whose roots is $a$. We evaluate it at some ...
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2answers
44 views

Composition series and its number determine a group?

By Jordan-Holder thm, it is known that every finite group has a unique composition series.(Here, unique means that there is only one kinds of such series.) And it is known also that composition ...
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452 views

Is ideal an “anti-field”?

I am comparing theorems on normal subgroup and ideal from Fraleigh's, and come to this strange intuition. I hope my conclusion does not screw up, I hope I won't get ridiculed: Theorem 15.18: $M$is ...
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2answers
47 views

Ideals of non semi-simple group rings.

I worked for a long time on complex group rings and complex twisted group rings. In those cases the algebra is semi-simple and its structure is well understood from the decomposition to irreducible ...
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65 views

Sets with $n$ prime numbers

Let $n>2$ a natural number. We define the following sets: $$S=\{1 \leq a \leq n : (a,n)=1, a^{n-1} \not\equiv 1\pmod n\} \\ T=\{1 \leq b \leq n : (b,n)=1, b^{n-1} \equiv 1 \pmod n\}$$ ...
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2answers
41 views

Question about a passage in Fulton and Harris

So I was reading the first chapter of Fulton and Harris and they are determining the representations of $S_3$. I came along this passage and had some questions What do they mean when they say "the ...
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1answer
64 views

Factoring $x^{15}−1$ into irreducible polynomials over $\mathrm{GF}(2)$

Factorize $x^{15}−1$ into irreducible polynomials over $\mathrm{GF}(2)$ The answer is $$(x+1)(x^2+x+1)(x^4+x+1)(x^4+x^3+1)(x^4+x^3+x^2+x+1)$$ but how would I find this? Please help.
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1answer
26 views

What is the name for this $R$-module? [on hold]

If $M$ is a $R$-module such that for all $x,y$ in $R$ and $m$ in $M$ then $x.y.m=0$ ... then: how is called $M$?
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129 views

Show that $t^n-1 \mid t^m-1 \Leftrightarrow n\mid m$

I want to prove the following lemma: $t^n-1$ divides $t^m-1$ in $F[t, t^{-1}]$ if and only if $n$ divides $m$ in $\mathbb{Z}$. I have done the following: $\Leftarrow $ : $n\mid m \Rightarrow ...
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Splitting even degree polynomials

I have an octic equation (degree $8$) and a sextic equation (degree $6$) in $\Bbb Z[x]$ with very large coefficients (size several hundred bits) that I know splits into two quartics and two cubics ...
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1answer
99 views

Solve $x^3+x+3=0$ by Galois's theory

Solve with radicals the following equation $x^3+x+3=0$, using Galois Theory. I'm just starting learning this and I do not have many ideas.
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2answers
33 views

Ring homomorphism and ideal that contains the kernel [on hold]

If $f:R\rightarrow S$ is a ring homomorphism and $I$ ia an ideal of $R$ such that $ker(f) \subseteq I$ then $f^{-1}(f(I))=I$ We know that $I\subseteq f^{-1}(f(I))$ but how can I use that $ker(f) ...
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1answer
40 views

A symmetric group question. [on hold]

Determine the integers $n$ such that there is a Surjective homomorphism from the symmetric group $S_n$ to $S_{n-1}$. It is a question from Artin's book. Exercise 7.5.8
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Kernel of ring homomorphism

Let $\phi: R \to R'$ be a ring isomorphism and $I$ an ideal of $R$. Define $\phi(I)=\{\phi(i): i \in I\}$. Show that $\frac RI \cong \frac {R'}{\phi(I)}$. To use the first isomorphism theorem, ...
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1answer
44 views

For each $n >2$ find all $m \in \mathbb N$ such that there exist a group $G$ of order $m$ and a surjective homomorphism $f: A_n \rightarrow G$

For each $n >2$ find all $m \in \mathbb N$ such that there exist a group $G$ of order $m$ and a surjective homomorphism $f: A_n \rightarrow G$< I tried to use the theorem of isomorphism, and a ...
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1answer
36 views

A $\mathbb Z/p\mathbb Z[G]$ submodule with no complement

Let $G$ be a group acting on a set $X$ of size $n$. Suppose $G$ acts doubly transitively. If $p$ is a prime, this naturally gives a permutation representation on the vector space over $\mathbb ...
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2answers
54 views

If $G = S_5$ and $H = \{g \in G \mid g^{5} = e\}$ how could I determine and prove whether or not $H$ is a subgroup of $G$?

I think that the this group contains the 5 element cycles and the identity e but overall I'm not sure how to prove that the product of the 2 members of H is also a 5 cycle or e.
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3answers
68 views

Why does $\sqrt{6} + \sqrt{10} + \sqrt{15}$ have four conjugates?

I am having trouble understanding how algebraic number $\sqrt{6} + \sqrt{10} + \sqrt{15}$ has four conjugates. Minimal polynomial is $x^4-62 x^2-240 x-239$ according to Wolfram Alpha. Factorized: ...
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80 views

Is the ring of holomorphic functions on $S^1$ Noetherian?

Let $S^1={\{ z \in \Bbb{C} : |z|=1 \}}$ be the unit circle. Let $R= \mathcal{H}(S^1)$ be the ring of holomorphic functions on $S^1$, i.e. the ring of functions $f: S^1 \longrightarrow \Bbb{C}$ which ...
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1answer
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Showing a nonabelian group of order 21 has an automorphism that is not inner.

I've seen at least 3 proofs of this on here, but most involved techniques I don't think I'm comfortable with, so I wanted to see if this one works: Since $21=3\cdot 7$, up to isomorphism there's only ...
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29 views

Groups,subgroups, normal subgroups [on hold]

Let $G$ be the group of all $2\times 2$ real matrices $\left( \begin{array}{ccc} a & b \\ 0 & d \end{array} \right) $ under matrix multiplication where $a,d\neq 0$. If $N=\left\{ \left( ...
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A problem about isomorphism in module theory

For a sequence of $R$-modules like this:$A \overset{f}{\rightarrow} B \overset{g}{\rightarrow}C$ such that $\mathrm{Im} f \subset\ker g$ and $B/\mathrm{Im}f \cong B/\ker g$. Then $\mathrm{Im}f=\ker ...
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1answer
45 views

Prove that a morphism $\alpha$ of $Fun(\mathcal{A},\mathcal{B})$ is an isomorphism iff each component $\alpha_A$, is an isomorphism in $\mathcal{B}$

I'm a computational engineer starting with a course of Introduction to Category Theory, and perhaps is extremely basic what I'm asking but I'm trying to learn how to make proofs in category theory ...
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1answer
33 views

Properties of isomorphism in module theory [on hold]

I have two exercises, but I can't solve them: a. If $X, A, B $ are $R$-modules with $A \subset B \subset X $, prove that if $X/A \cong X/B$ then $A=B$ b. If $X/A \cong X$ then $A=0$
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38 views

Inducing representations from a subgroup of finite index.

Let $G$ be a group and $H$ a subgroup of finite index. Let $(\sigma , W) $ be a irreducible representation of $H$ (which need not be finite dimensional). 1) When can we extend this representation of ...
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1answer
27 views

I have to show no proper intermediate fields exist between $Z_2$ and $GF(2^3)$ [duplicate]

I have to show no proper intermediate fields exist between $Z_2$ and its overfield $GF(2^3)$, Can any one help?
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1answer
34 views

Prove the theorem of ideal (about g.c.d)

If $p_1,\ldots,p_n$ are polynomials over a field $F$, not all of which are $0$, there is a unique monic polynomial $d$ in $F[x]$ such that (a) $d$ is in the ideal generated by $p_1, \ldots, ...
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38 views

Irreducible component of a scheme over a dvr

Let $\mathcal M$ be a (reduced) quasi-projective scheme over a dvr (of mixed caracteristics), $R$. Suppose that the generic fiber $\mathcal M_{\eta_R}$ is (nonempty) smooth and irreducible of ...
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2answers
37 views

How do I construct a nontrivial linear representation of the group $G$=S' in $R^3$

$S'$ is defined as the unit circle. The product is defined as the sum of angles. How do I construct a linear representation in general? I don't know how to begin with this problem. Some ...
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44 views

Number of all possible groups of given order [duplicate]

Suppose $n=18$, then all possible groups of order $18$ is $5$. Among them $2$ are abelian and $3$ are non-abelian. Let $n$ be a natural number. How can I determine the number of all possible ...