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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.

0
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0answers
6 views

Isomorphism problem for the center of modular group algebras

Let $p$ be a prime number, $G,H$ a finite p-Group and $K$ a finite field with $\operatorname{char}(K)=p$. It is well-known that the group $1+\operatorname{rad}(KG)$ is a p-group containing $G$. My ...
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0answers
6 views

A question to the ascending central chain in modular group algebras

Let $p$ be a prime number, $G$ a finite p-Group and $K$ a finite field with $char(K)=p$. It is well-known that the group $1+rad(KG)$ is a p-group containing $G$. Let us focus on the sequence $G\cdot ...
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0answers
9 views

Defect of subnormality in modular group algebras

Let $p$ be a prime number, $G$ a finite p-Group and $K$ a finite field with $\operatorname{char}(K)=p$. It is well-known that the group $1+\operatorname{rad}(KG)$ is a p-group containing $G$. $G$ is ...
8
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1answer
44 views

Is this proof of $H\le G, [G:H] =2 \implies a^2\in H \forall a\in G$ correct?

If $H$ is a subgroup of a group $G$ and the index (number of right cosets) of $H$ is $2$, then $a^2 \in H$ for all $a\in G$. My attempt: if $a\in H$ then $a^2\in H$ directly. If $a\notin H$ and $a^2\...
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0answers
29 views

Definition of action of Lie algebra of an algbraic group

Here is the context of my interrogation : Let $G$ be an affine algebraic group over $\mathbb{C}$ acting rationally on an affine variety $X$ over $\mathbb{C}$. This induces an action of $G$ on $\...
1
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1answer
34 views

How do I find the kernel of a given group homomorphism?

I am given a group $H$ and two normal subgroups $A$ and $B$ and $H/A$ and $H/B$ are two quotient groups. A homomorphism is defined as follows: $\phi: H \to H/A \times H/B; \phi(h) = (hA,hB)$. How do ...
2
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2answers
166 views

Simplicity of the roots of a minimal polynomial

Let $L/K$ be a finite field extension, and let $\mu_{\alpha,K}\in K[X]$ be the minimal polynomial of $\alpha\in L$. One can easily see that $\alpha$ is a simple root of $\mu_{\alpha,K}$. Indeed, if ...
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0answers
62 views

Group of units of a field

So I'm a bit confused with this... I'm learning some field theory and I've just learnt about groups of units. With rings this makes sense. However say $F$ is a field then isn't every element in $F$ ...
0
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1answer
44 views

Property of a category transfers to a subcategory

Is there a category theory condition such that we can say when a property of a category also holds for it's sub-category? For example, in the catgeory of groups $f$ being bijective implies it is an ...
2
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1answer
62 views

Functor of points of the completion of a ring

Let $R$ be a ring, with ideal $I$, and let $\widehat{R}_I$ be the completion $\varprojlim R / I^n$ of $R$. Can I somehow describe the functor $\mathrm{Hom}(\widehat{R}_I,?) : Rings \to Sets$ in an ...
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1answer
37 views

If $G$ is $s$-step nilpotent and $n \in \mathbb N$, then $(G_s)^{n^s} \subset (G^n)_s \subset (G_s)^n$

Define the following "commutators" recursively: $[g,h] = g^{-1}h^{-1}gh$ $[g_1, \dots, g_m] = [g_1, [g_2, \dots, g_m]]$ for all $m \geq 3$ Let $G_i$ be the lower central series of a group, $G$. ...
2
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0answers
21 views

Proof that orthogonal conjugation is self-inverse

I am trying to prove that given a cyclic gropu $G$ and a subgroup $H$ then $(H^\bot)^\bot = H$, where $H^\bot = \{ \alpha \in G | \chi_\alpha(x) = 1 \forall x \in H\}$ and $\chi_\alpha$ are fourier ...
2
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0answers
24 views

Prime property in noncommutative rings without identity

Let $R$ be a ring (without assuming identity or commutativity), and $P$ a proper ideal of $R$. Show that the following are equivalent: (a) For ideals $A,B$: $AB\subseteq P$ implies $A\subseteq P$ ...
-1
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1answer
26 views

If $K$ is a field then $\frac{K[x_1,x_2,…,x_n]}{(x_1-\alpha_1,…,x_i-\alpha_i)}\cong K[x_{i+1},…,x_n]$

If $K$ is a field and $\alpha_1, \alpha_2, ..., \alpha_p \in K$ then $$\frac{K[x_1,x_2,...,x_n]}{(x_1-\alpha_1,...,x_i-\alpha_i)}\cong K[x_{i+1},...,x_n] $$ My Attempt I simply tried to use Hilbert'...
2
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1answer
75 views

Prove that subgroup of all elements of finite order in group $(\mathbb{C} \setminus \{0\}, \cdot)$ is isomorphic to $\mathbb{Q}/\mathbb{Z}$

Let $H$ be the subgroup of all elements of finite order in group $\left(\mathbb{C} \setminus \{0\}, \cdot \right)$. Prove that $H$ is isomorphic to $\mathbb{Q}/\mathbb{Z}$, where $\mathbb{Q}$ and $\...
2
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1answer
41 views

Prove that these two conditions are equivalent

Let $m, n \in \mathbb{N}$. Prove that these two conditions are equivalent: $m, n$ are coprime; for any group $G$, for any subgroup $A \subseteq G$ of order $m$ and for any subgroup $B \subseteq G$ of ...
0
votes
1answer
41 views

If $R$ is a PID then $\mathrm{Spec}(R)\!=\!\mathrm{Max}(R)\!\cup\!\{0\}$? [on hold]

Is the following statement true? If $R$ is a PID then $\mathrm{Spec}(R)\!=\!\mathrm{Max}(R)\!\cup\!\{0\}$ I know the reverse is false.
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1answer
32 views

Do we lose much if we take isomorphism as equality in a purely algebraic context?

In algebra, we sometimes end up with a sentence relating two objets via an isomorphism. As algebra is kind of spiritual behavior of objects, (we almost never speak about the object, but we speak about ...
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0answers
48 views

Circle group $S^1$

Can someone describe the circle group $S^1$ in a easily understandable way? Are elements in $S^1$ in the form of $e^{2i\theta\pi}$? What does this look like pictorially? Doesn't necessarily need a ...
0
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1answer
32 views

In general, what's the relation between $\text{Ext}^n_R(A,B)$ and $\text{Ext}^n_R(B,A)$ (if any)?

Here $A,B$ are arbitrary $R$-modules. Similarly, what's the relation between $\text{Tor}^n_R(A,B)$ and $\text{Tor}^n_R(B,A)$? I am on 17.1 Dummit and Foote and couldn't find any remarks on this.
3
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1answer
42 views

Does the category of $(R,S)$-bimodules contain a free object on one generator?

I believe that in the category of $(R,S)$-bimodules where $R$ and $S$ are rings with identity, then ${}_R R \otimes_\mathbb{Z} S_S$ is a free object on one generator in this category. But, if $R, S$ ...
2
votes
2answers
35 views

Identifying simple tensors.

Let $S$ be a domain. I want to determine whether or not, every element of $\text{Frac(S)}\otimes_S M$ is a simple tensor, where $M$ is any $S$-module. I couldn't produce a tensor that is not pure in ...
0
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1answer
28 views

Find all subfields of $\mathbb{Q}(\mu_{24})$

Problem: Let $\mu_{24} \in\mathbb{C}$ be a primitive 24'th root of unity and let $L = \mathbb{Q}(\mu_{24})$ be the 24'th cyclotomic extension of $\mathbb{Q}$. List all subfields of $L$ in the form $\...
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votes
0answers
47 views

what is the area of 7 squares each of area 5.29 sq units taking significant figures into consideration? [closed]

it is question of jee main 2019. what is the area of 7 squares each of area 5.29 sq units taking significant figures into consideration ? what is the area of 7 squares each of area 5.29 sq units ...
6
votes
2answers
71 views

Prove that $[\mathbf{Q}(\sqrt{1+i},\sqrt{2}):\mathbf{Q}]=8$.

I am trying to calculate the Galois group of the polynomial $f=X^4-2X^2+2$. $f$ is Eisenstein with $p=2$, so irreducible over $\mathbf{Q}$. I calculated the zeros to be $\alpha_1=\sqrt{1+i},\alpha_2=\...
1
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1answer
24 views

Is $X\cong\textrm{Ker}(f)\oplus\textrm{Im}(f)$ for a module homomorphism $f:X\to Y$ with semisimple domain?

Let $f:X\to Y$ be a module homomorphism with semisimple domain. Does \begin{equation*} X\cong\textrm{Ker}(f)\oplus\textrm{Im}(f) \end{equation*} hold true that? (In my previous question it was kindly ...
1
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2answers
80 views

Is it $Z(\operatorname{Aut}(G)) \cap \operatorname{Inn}(G) \cong H/Z(G)$ for some $H \le G$?

Could you check if this proof is correct, please? (I'm not even sure about the result itself, whence the title.) Proposition. Let $G$ be a group. Then: $$Z(\operatorname{Aut}(G)) \cap \operatorname{...
1
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2answers
42 views

Sum of signature of elements of $S_n$ is $0$

I saw in a proof that for each $n > 1$, the symmetric group $S_n$ satisfies $$\sum_{g\in S_n} \varepsilon(g) =0,$$ where $\varepsilon$ is the signature. Is that true? I checked it is true ...
1
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0answers
18 views

Local Representation Theory Alperin Ex 1.4.

Let $A$ be a finite dimensional $k$-algebra where $k$ is an algebraically closed field. Let $U$ be an $A$-module of Loewy length $s$. We wish to show $$rad^i(U) \subseteq soc^{s-i}(U)$$ for $0 \leq ...
1
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1answer
35 views

Intersection of finite Galois extensions is Galois

If K and L are finite Galois extensions of F, then show that the intersection of K and L is Galois over F. I was trying to use Galois group of the intersection and show that if the fixed field is NOT ...
3
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1answer
42 views

$\sigma \in \mathrm{Gal}(K/k), \sigma \alpha \ne \alpha$, but why is $\alpha \in k$?

Suppose that $k$ contains $\zeta$, a primitive $p$-th root of unity where $p$ is prime, and that $K$ is Galois over $k$ with $[K : k]=p$; and write $G=\operatorname{Gal}(K / k) \approx C_p$. Show ...
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0answers
35 views

Computing the kernel of $A[t] \rightarrow K : t \mapsto \frac{a}{b}$.

I have a complete local Noetherian normal domain $A$ (commutative with 1) with fraction field $K$. I was trying to compute the pullback of the prime ideal $(t - \frac{a}{b}) \subset K[t]$ to $A[t]$. ...
3
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2answers
193 views

Left action of a group on permutation representation

I am studying my course on permutation representation, and I am stuck at understanding the left action of a finite group on the permutation representation $F(X,\Bbb C)$. In my course it is given for $(...
0
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1answer
34 views

Subgroup of $S_4$ generated by $\{(123), (12)(34)\}$

I refer to the following problem. Determine the subgroup of $S_4$ generated by $\{(123), (12)(34)\}$. In his solution to the problem the author makes the following claim: As $(123) \in A_4$ and ...
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2answers
70 views

“For all X, X =A iff X=B, therefore the A = B”. Is this logically correct?

Suppose I want to prove ( in elementary arithmetics, or, maybe, in an abstract additive group) that : the additive inverse of (a+b) = -b + -a May I proceed as follows? Suppose X = - ( a+b). Now,...
2
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1answer
38 views

Conjugacy classes in space of trace zero 2*2 matrices

I'm trying to find the orbits when $SL_2$ operates by conjugation on $\mathfrak{sl}_2=Lie(SL_2)=\{A|\operatorname{tr} A=0\}$. I have tried to write $X\in sl_2$ and corresponding $AXA^{-1}$ for random ...
2
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0answers
35 views

What does vanishing of higher direct images of the structure sheaf tell us?

Let $f:X\to Y$ be a morphism of schemes over $k$. I am wondering about, what geometric consequences $R^qf_*O_X=0$ for $q\geq k$ does have. I saw vanishing of higher direct images used in some proofs ...
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2answers
36 views

How to find number of abelian subgroups of diheral group? [closed]

How to find number of abelian subgroups of diheral group $D_n $? Attempt: I have counter-examples for $n=1,2$ so I know that it isn't true for $n<3$. Is it true for $n\ge 3$? How do you know this?...
0
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1answer
50 views

Find prime ideals of Quotient Ring

Let $R$ be the ring $\Bbb C[x]/(x^2+1)$. Pick the correct statements from below: $\dim_\Bbb C R=3$. $R$ has exactly two prime ideals. $R$ is a UFD. $(x)$ is a maximal ideal of $...
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votes
1answer
16 views

Finding order and cosets of subgroup [closed]

For: Let $P$ be the pairs $(a,b)$ where $a \in \Bbb Z_4$, and $b \in \Bbb Z_2$ An operation, $*$, is defined by: $$(a,b)*(c,d)=(a+c \pmod 4, b+d \pmod 2)$$ for all $(a,c),(b,d)\in P$ H = ...
1
vote
2answers
83 views

Showing $\mathbb Z_4 \times \mathbb Z_2$ to be a group

I've been posed the question: Let $P$ be the pairs $(a,b)$ where $a \in \Bbb Z_4$, and $b \in \Bbb Z_2$ An operation, $*$, is defined by: $$(a,b)*(c,d)=(a+c \pmod 4, b+d \pmod 2)$$ for all $(...
0
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1answer
29 views

Quintic polynomial with three real roots

I want to get a quintic polynomial $f(X) \in \mathbb{Q}[X]$ whose Galois group $\mathrm{Gal}(L/\mathbb{Q}) \cong S_5$ where $L$ is the splitting field of $f(X)$. One of strategies to get it is ...
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0answers
12 views

Construct parity check matrix of binary Goppa Code

I am working out on a problem given in the book Theory of Error-correcting codes by MacWilliams and Sloane. The problem is to construct a parity check matrix for a classical binary $[8,2,5]$ Goppa ...
4
votes
1answer
101 views

Is $R[x]/(x^2)$ isomorphic to $R[x]/(x^3)$?

I guess no, because $x^3$ is not divisible by $x^2$, but I am not sure whether it is right to think in this way or not. Remarks: R[x] denotes the set of all polynomials with coefficients in R. $R[x]/...
0
votes
1answer
39 views

Finding primitive element

Let $a_1,\ldots, a_n$ be integers such that $[\mathbb{Q}(\sqrt{a_1},\ldots, \sqrt{a_n}):\mathbb{Q}]=2^n$. I want to show that $\sqrt{a_1}+\cdots+\sqrt{a_n}$ is a primitive element of $\mathbb{Q}(\sqrt{...
2
votes
1answer
48 views

When the characteristic of the field is not two, we can always find an orthogonal basis.

Let $ V $ be a $n$-dimensional vector space over the finite field $\mathbb F_q$, with $ \operatorname{Char}\mathbb F_q\ne 2 $. Show that for every symmetric bilinear form $B(\cdot,\cdot)$ on $V$, ...
1
vote
2answers
44 views

How is this classical group $\textit{compact}$? [duplicate]

Let $O(n)$ be the group of orthogonal $n \times n$ matrices. Apparently this is a "compact classical group" but I have trouble seeing that it is compact. The topology is the topology is inherits from $...
0
votes
2answers
66 views

Mapping from $\mathbb Z/2\mathbb Z \times \mathbb Z/3\mathbb Z \times \dots\times \mathbb Z/p_n\mathbb Z$ to $\mathbb Z/p_n\#\mathbb Z$.

I know $\mathbb Z/2\mathbb Z \times \mathbb Z/3\mathbb Z \times \dots\times \mathbb Z/p_n\mathbb Z$ is isomorphic $\mathbb Z/p_n\#\mathbb Z$ (where $p_n\#$ is the primorial of primes up to $p_n$) by ...
2
votes
1answer
84 views

Splitting field of $\sqrt{\vphantom{\sum}1+{\sqrt2}}$ and Galois group

Let $\alpha= \sqrt{\vphantom{\sum}1+{\sqrt2}}$. (a) Let $p(x)$ be the minimal polynomial of $\alpha$. Find $p(x)$. Let K be the splitting field of $p(x)$ (b)Let $E= \mathbb{Q}(i, \sqrt2)$ Show that $...
-2
votes
0answers
28 views

Showing that rigid rotations are distinct. [on hold]

Let $T_{\theta}: S^1 = \mathbb{R}/\mathbb{Z} \rightarrow S^1 = \mathbb{R}/\mathbb{Z} \\$ $x \bmod 1 \mapsto x + \theta \bmod 1$. Let $T$ be rigid. Let $\theta$ be irrational. Prove that $0,\ T_{\...