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

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structure of the unit group modulo a prime power

Let $p$ be an odd prime and $m \geq 2$. Show that $\mathbb{Z}_{p^{m-1}} \times (\mathbb{Z}_{p})^{\times} = (\mathbb{Z}_{p^{m}})^{\times}$ I am having trouble coming up with the correct map that ...
3
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1answer
115 views

Which groups have the property that $HK$ is a subgroup for all noncyclic subgroups $H$ and $K$?

At first I wondered Which groups $G$ have the property that $HK$ is a subgroup for all subgroups $H,K<G$? If a finite group $G$ has this property, I think that $G$ ought to be nilpotent. (I ...
4
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1answer
301 views

Prove $G/\ker \phi \times \ker \phi \cong G$

If $G, H$ are groups and $\phi : G \to H$ is a homomorphism, is it true that $G/\ker \phi \times \ker \phi \cong G$? I am pretty sure this is right, but I can't remember how to prove it. We can ...
2
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1answer
137 views

Find the torsion subgroup of $\mathbb{Z}\times (\mathbb{Z}/n\mathbb{Z})$

Let $G$ be an abelian group. $\left \{ g\in G||g|< \infty \right \}$ is a subgroup of $G$, called the torsion subgroup of $G$. Fix some $n\in \mathbb{Z}$ with $n>1$. The question is to find ...
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2answers
508 views

On Conjugacy Classes of Alternating Group $A_n$

In Dummit & Foote, page 131 Let $K$ be a conjugacy class and suppose that $K$ is subset of $A_n$ . Show that if $\sigma$ belongs to $S_n$ then , $\sigma$ does not commute with any ...
4
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1answer
659 views

Specific proof that any finitely generated $R$-module over a Noetherian ring is Noetherian.

I have seen a handful of proofs that any finitely generated module over a Noetherian ring is again Noetherian. I'm specifically trying to understand the following proof idea. It goes as this: Observe ...
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1answer
108 views

Separability of a field Extension.

Let $f=x^n-1$, and $L$ be a splitting field of $f$ over $K$ . Basically my question is to show that the extension $L|K$ is separable . Here is what i have been thinking , If char $K =0$ then its ...
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3answers
182 views

Finding the ideals in a ring of fractions

I am dealing with the ring $$R=\left\{\frac{a}{b} \mid a,b\in\mathbb{Z}\mbox{, $b$ is not divisible by 3}\right\}$$ with addition and multiplication as defined in $\mathbb{Q}$ and I'm trying to find ...
2
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2answers
48 views

dual representation — show that it actually is a linear representation in the dual space

Let $\rho : G \to \mathrm{GL}(V)$ be a linear representation in $V$. Show that $\rho^* : G \to \mathrm{GL}(V^*)$ with $\rho^*(g)(f) = f \circ \rho(g^{-1})$ is a linear representation on $V^*$. So ...
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2answers
663 views

Subgroups of order $p$ and $p^{n-1}$ in a group of order $p^n$.

I have a group $G$ of order $p^n$ for $n \ge 1$ and $p$ a prime. I am looking for two specific subgroups within $G$: one of order $p$ and one of order $p^{n-1}$. I don't think I would use the Sylow ...
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129 views

Easier way to show irreducible?

I want to show that the following polynomial is irreducible in $\mathbb{Z}[X]$: $$f(X) = X^4 -4X^3 -4X^2 + 16X - 8.$$ I thought it was irreducible mod 3 but someone pointed out that was wrong. Any ...
4
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2answers
176 views

How to show $e^{2 \pi i \theta}$ is not algebraic.

I was wondering if someone could possibly help me figure out how to show $e^{2 \pi i \theta}$ is not algebraic when $\theta$ is irrational. Thanks!
3
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1answer
99 views

Commutative $ \mathbb{C} $-algebra with involution.

The following is a problem in the Professor’s lecture notes; I don’t know how to prove it: A complex $ ^{*} $-algebra $ A $ is commutative if and only if the set of self-adjoint elements of $ A $ ...
5
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3answers
200 views

The free group $F_2$ contains $F_k$

I want to prove the following: the free group $F_2$ contains the free group $F_k$ for every $k \geq 3$. I am wondering whether the following line of reasoning is correct or not: Suppose that $\lbrace ...
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2answers
187 views

How can we know arithmetical axioms are consistent?

If we assume both distributivity and the opposite of the law of signs (ie, that $-1\times-1 = -1$) for the relative integers, then we can derive that two different numbers are actually equal. ...
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2answers
163 views

There is no non-trivial homomorphism $\mathbb{Q}\rightarrow S_3$

I want to show that the following problem is true : There is no non-trivial homomorphism $\mathbb{Q}\rightarrow S_3$ Please help me to show it. Thanks.
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441 views

A multiple choice question on finite group [closed]

Let $G$ be a finite group such that $Z(G)=1$. Let there exist $m$ such that $G$ has a unique element of order $m$. Which of the following statements is true? (a) $m=1$ (b) $m$ is prime (c) ...
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1answer
164 views

Normal subsets of a Sylow p subgroup are conjugate if and only if they are $N_G(P)$ conjugate.

The following is a question from Dummit & Foote. Prove that if $U$ and $W$ are normal subsets of a Sylow $p$-subgroup $P$ of a finite group $G$ then $U$ is $G$-conjugate to $W$ if and only ...
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2answers
478 views

Structure of Finite Commutative Rings

Is every finite commutative ring $A$ a direct product of finite algebras over $\mathbb Z/p^n$?
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188 views

If $a$ is a unit and $b$ is a zero divisor in a ring $R$, then $ab$ is a zero divisor

I know that $ax=1$ has a nonzero solution, $bx = 0$ has a nonzero solution. I am of course trying to show that $(ab)(x) = 0$ has a nonzero solution.
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1answer
310 views

Subgroup of a free group of finite index

I'm trying to prove the following: if $F$ is a free group and $H$ is a subgroup of $F$ such that the index $[F:H]$ is finite, then $H \cap K \neq 1$ for every nontrivial subgroup $K$ of $F$. I don't ...
3
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3answers
358 views

Field extensions that are not normal

I am trying to come up with field extensions $M : L : K$ such that none of the three extensions $M:L, L:K, M:K$ are normal. So far, I have tried letting $K = \mathbb{Q}, L = \mathbb{Q}(\sqrt[3]{2})$. ...
2
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1answer
80 views

Bijection between a variety $X$ and $\hom(k[X], k)$

I have seen a theorem some time ago, and I don't remember the exact assumptions, so there may be some mistakes. The statement is the following: If $X$ is an affine variety over an algebraically ...
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1answer
192 views

Can extending a finite ground field make modules isomorphic?

$\def\Hom{\mathrm{Hom}}$Let $k$ be a field, $A$ a $k$-algebra and let $M$ and $N$ be $A$-modules, finite dimensional over $k$. Let $K$ be an extension of $k$, so $A \otimes K$ is a $K$-algebra and $M ...
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1answer
134 views

Boolean ring. Representation as direct product?

I am reading Atiyah, Macdonald's book "Commutative algebra". There is an exercise, which states that that if a ring with identity has idempotent element $\ne 0,1$, then the ring is a direct product ...
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1answer
201 views

Questions about Hamel basis for ${\mathbb R}$ over ${\mathbb Q}$

If I understand the whole Hamel basis idea correctly, there exists one such basis $B = \{v_\alpha\}_{\alpha \in I}$ for ${\mathbb R}$ (herein construed as a vector space of ${\mathbb Q}$), such that ...
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3answers
522 views

Ring of order $p^2$ is commutative.

I would like to show that ring of order $p^2$ is commutative. Taking $G=(R, +)$ as group, we have two possible isomorphism classes $\mathbb Z /p^2\mathbb Z$ and $\mathbb Z/ p\mathbb Z \times \mathbb ...
2
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1answer
138 views

What are some slick ways to prove that a presentation is actually isomorphic to a given group?

Let's say I have a particular finite presentation and want to show it's actually a presentation for the group I claim it's a presentation for. That group might be specified, say, by a linear or ...
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2answers
190 views

Do these two observations suffice to show that a finite boolean ring must be of the form $\mathbb{Z}_2\times\cdots\times\mathbb{Z}_2$?

Question: If $R$ is a finite boolean ring, then show $R \cong \mathbb Z_2 \times \mathbb Z_2 \times \cdots\times \mathbb Z_2$. I know that $\mathrm{char}(R) =2$ $R$ has $2^k$ elements for some ...
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1answer
64 views

whether the map is surjective?

question : the natural map defined by $\phi :\mathbb Z[x] \rightarrow$ $\mathbb Z[x]/<2> \times$ $ \mathbb Z[x]/<x> $ is not surjective . ans $\phi(f(x))=(\pi_1(f(x)),\pi_2(f(x)))$ ...
2
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3answers
189 views

Group Automorphisms of fields

Let $k$ be an algebraically closed field and $(k,+)$ be the additive group. I have read somewhere that the group automorphisms of $(k,+)$ are exactly the multiplications by non-zero elements of $k$. I ...
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Ways of finding primitive element of separable extension.

Consider the field extension $L=\mathbb Q (\sqrt[4] 2 ,i)$ over $\mathbb Q$. This extension is separable as we know over a field of characterstic $0$. Now according to the primitive element theorem ...
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votes
3answers
319 views

How can the real numbers be a field if $0$ has no inverse?

I'm reading a linear algebra book (Linear Algebra by Georgi E. Shilov, Dover Books) and the very start of the book discusses fields. 9 field axioms discussing addition and multiplication are given ...
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1answer
134 views

Frattini subgroup of $\operatorname{Hol}(\mathbb Q)$

Let $\mathbb Q$ be rational number under addtion. Is Frattini subgroup of $\operatorname{Hol}(\mathbb Q)$ trivial where $\operatorname{Hol}(\mathbb Q)=\mathbb Q \rtimes \operatorname{Aut}(\mathbb Q) ...
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Pairing two algebra structures of the same type and result the same type of algebra structure

I found one can pair two algebra structure and result the same type of algebraic structure. For example, if $(S,+)$ and $(T,+')$ are semigroup. Let $M=S\times T$ and $(a,b)\oplus(c,d)$ is defined as ...
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2answers
133 views

Powering map is surjective when the power is relatively prime to the order of the group

question: let $|G| = n$ and $(k,n)=1 $ where $k $ is an integer and $\phi :G \rightarrow G$ defined by $\phi(g) =g^k$ then to show $\phi$ is surjective. ans: $(n,k)=1 \implies an +bk=1$ for some ...
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2answers
308 views

finite odd order abelian group property

question: let $|G|=odd $ where $G$ is a finite commutative group then to show every element of $G$ is a square. ans 1> to show that $∀g∈G,∃g_1∈G,g=g_1^2$. let $g \in G$ then $|g|$ $\big |$ $|G| ...
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6answers
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Is $\mathbf{C}$ the algebraic closure of any field other than $\mathbf{R}$?

It seems to me (intuitively) that there should be no other fields whose algebraic closure is $\mathbf{C}$, even though I have no reason to believe it. The facts I've been using to formulate an ...
6
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3answers
450 views

Can every real polynomial be factored up to quadratic factors?

I know that you can't factor a real polynomial into $\Pi_{i=1}^N(x-a_i)$ in general. But is it possible to factor every real finite polynomial into this form: $(\Pi_{i=1}^N a_ix^2 + b_ix + c_i) ...
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4answers
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Show that a nonempty set of integers that is closed under subtraction must also be closed under addition

So this is what I have so far: Let X be a nonempty set of integers Let $a,b\in X$ and we need to show that $a+b\in X$ Because $b\in X$ and X is closed under subtraction, than $b-b\in X$ Once ...
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1answer
387 views

Artin-Wedderburn decomposition of a particular group ring

I am trying to do a question from an algebra qualifying exam: Decompose the group ring $\mathbb{F}_5[S_3]$ as a product of simple rings. By Maschke's theorem since $\mathrm{char}(\mathbb{F}_5) ...
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Good reference for co-groups, perspective of co-algebra applications

There are lot of applications of state transition systems STS (computer science, planning problems in robotics and so on) and lot of algorithms are devised, but the mathematical background for STS is ...
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2answers
355 views

Size of conjugacy classes in $GL(4,2)$

I'm asked to find out all of the conjugacy classes, their order and their size for $GL(4,2)$. Finding representatives is possible by looking for all the rational canonical forms over the field and ...
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1answer
2k views

The (Jacobson) radical of modules over commutative rings

Let $M$ be a module over a commutative ring $R$. Let $\Omega$ be the set of all maximal ideals of $R$. Prove that $\operatorname{Rad}(M)=\bigcap_{\mathfrak m\in \Omega}\mathfrak mM$, where ...
6
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1answer
313 views

Let $A,B$ be two subsets of a finite group $G$. If $|A|+|B|>|G|$, show that $G=AB$

Let $A,B$ be two subsets of a finite group $G$. If $|A|+|B|>|G|$, show that $G=AB$. My attempt is : Since $|A|+|B|>|G|$, there exists one common element in both sets $A$ and $B$, say $g$. Then ...
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147 views

Injective group homomorphism from $\mathbb Z_d$ to $S_n$

For $d < n$, define an injective function $\mathbb Z_d \rightarrow S_n$ preserving the operation, that is, such that the sum of equivalence classes in $\mathbb Z_d$ corresponds to the product of ...
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1answer
93 views

Semigroup homomorphism and the relation $\mathcal{R}$

Let $S$ be a semigroup and for $a\in S$ let $$aS = \{as : s \in S \}\text{,}\;\;\;aS^1 = aS \cup \{a\}\text{.}$$The relation $\mathcal{R}$ on a semigroup $S$ is defined by the rule: $$a\;\mathcal{R}\; ...
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1answer
58 views

If $[G:H]$ is a prime $p$ and $H \triangleleft G$, show that there exists an element $a \in G$ such that $G=\cup_{i=0}^{p-1}a^iH$

Let $H$ be a subgroup of $G$.If $[G:H]$ is a prime $p$ and $H \triangleleft G$, show that there exists an element $a \in G$ such that $G=\cup_{i=0}^{p-1}a^iH$. I have no idea on how to start. Anyone ...
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3answers
392 views

Symmetry group of a triangular lattice

The question is: Let $G$ be the group of symmetries of an equilateral triangular lattice $L$. Find the index in $G$ of the subgroup $T \cap G$. ($T$ is the group of translations) The solution ...
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190 views

An Algorithm to Find the Generators of the Radical of a Monomial Ideal

Working over $R=\mathbb{C}[x_1,...,x_n]$, I'm given a ring homomorphism with $i\in{1,...,n}$ and $t\in \mathbb{C}$. $\phi_{i,t}(x_j)=x_j$ for $j\neq i$ to themselves. From this I've proven that an ...