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.

learn more… | top users | synonyms (1)

0
votes
0answers
4 views

Extension of group

Let $ p = 2 $, and Let $ H $ be a subgroup of a direct product of copies of $ S_{3} $. Why $ H $ is an extension of a $ 3 $-group by a $ 2 $-group, and $ H/O_{p^{\prime}}(H) $ is a $ 2 $-group ?
2
votes
3answers
20 views

A more formal but intuitive understanding (on a definition) of a group action

We know that a symmetric group $S_n$ acts on the set $\{1, 2,\ldots, n\}$. The definition of an action of a group $G$ on a set $S$ is a function $G\times S\to S$ such that: 1) $e\ast s=s$ 2) ...
1
vote
0answers
16 views

Why $ G/F $ is a $ 2 $-group?

Let $ G $ be a finite soluble group and $ A $ is the unique minimal normal subgroup of $ G $. $ N = Fit(G) $. Every chief factor of $ G/A $ has order $ 4 $ or a prime. Let $ \vert A \vert = p^{\alpha} ...
1
vote
1answer
26 views

Find isomorphism between $S_3$ and $GL_2(F_2)$.

Find isomorphism between $S_3$ and $GL_2(F_2)$. proof: Let $A = \begin{pmatrix} a& b\\ c & d \end{pmatrix}$. Where $\det (A) \neq 0$. And recall $S_3$ is the permutation group with ...
1
vote
2answers
18 views

Is $\mathbb{Q}(\alpha_i^2)$ an intermediate field of $\mathbf{K}/\mathbb{Q}$, where $K$ is the splitting field of $x^3-2$ over $\mathbb{Q}$?

I've found the Galois group of $x^3-2$, isomorphic to $\mathbf{S}_3$. It has 6 subgroups (including the trivial subgroup and the group itself), and thus by the Galois Correspondence there should be 6 ...
0
votes
1answer
20 views

Question in line of proof for first isomorphism theorem

Let $\phi: G_1 \to G_2$ be a group homomorphism. Let $\ker \left({\phi}\right)$ be the kernel of $\phi$. Then: $\operatorname {Im} \left({\phi}\right) \cong G_1 / \ker ...
0
votes
0answers
27 views

In Z_437, calculate 30 circled division 29

I am having trouble with modular division, especially finding inverses. I know the answer is 212, but I was hoping someone could show me how to reach this answer. Other practice problems include (all ...
1
vote
1answer
26 views

Field homomorphism induces an isomorphism between their prime subfields

So the question is: Let $\sigma$: $F_1 \xrightarrow[]{} F_2$ be a homomorphism where $F_1$ and $F_2$ are fields. Show $\sigma$ induces an isomorphism between their prime subfields and, in ...
1
vote
2answers
63 views

What's the importance of proving that $0,1$ are unique?

I had a course in the construction of numbers last semester. I understand the potencial of most of the proofs, for example: I guess I can answer decently why commutativity is important. But when it ...
1
vote
0answers
24 views

Field of formal Laurent series over $F$.

Let $F$ be a field and let $K$ be the set of all functions $f\in F^\mathbb{Z}$ satisfying the condition that there exists an integer (perhaps negative) $n_f$ such that $f(i)=0$ for all $i<n_f$. ...
1
vote
2answers
32 views

Let $f:Z \times Z \to Z$ with $f(1,1)=2$ and $f(3,5)=6$. Estimate the $\ker f$ of $f$ and $f(0,5)$

Let $f:Z \times Z \to Z$ with $f(1,1)=2$ and $f(3,5)=6$. Estimate the $\ker f$ of $f$ and $f(0,5)$. I am trying to solve this but i need any ideas or hints to start,any help would be interesting.
-4
votes
0answers
35 views

Linear Algebra help needed [on hold]

It's been a while since I've taken linear algebra and I am trying to figure out the problem below. I am not sure how to start. Thank you for any and all help. I am not sure how to For any real ...
1
vote
0answers
28 views

Determining the kernel of a module homomorphism

Let $p$ be a prime and let $n$ be a positive integer such that $p^n > 2$. Set $R:= \mathbb{Z}_{p^n}$, that is, the residue ring with binary operations of addition and multiplication modulo $p^n$. ...
1
vote
1answer
23 views

Show $R \setminus S$ is a union of prime ideals

I'm stuck on the following question: Let $R$ be a commutative ring with $1$, and $S \subseteq R$ a saturated multiplicative set (that is, $1 \in S$ and $x, y \in S$ if and only if $xy \in S$). ...
0
votes
0answers
23 views

Hanging a painting with nails so that removing any subset of nails from a given collection makes painting fall, and subsets are minimal

So I'm aware of the result that for positive integers $k \leq n$ it's possible to hang a painting with $n$ nails, such that if any $k$ nails are removed then the painting falls, but never when $k-1$ ...
2
votes
1answer
18 views

$ G/N $ is is a subgroup of a direct product of copies of the cyclic group $ C_{p-1} $ if $ p>2 $

Let $ G $ is soluble group and $ A $ be a unique minimal normal subgroup of $ G $. Then $ A $ is a elementary abelian group of a prime power. Let every chief factor of $ G/A $ has order $ 4 $ or a ...
0
votes
2answers
21 views

How to show #Hom$(C_a, G)=\{x\in G: x^a=e\}$?

I am willing to establish that $$\#\text{Hom}(C_a, G)=\#\{x\in G: x^a=e\}$$ where $G$ is finite group of order $n$ and $C_a$ is cyclic group of order $a$. I started like this: By first isomorphism ...
1
vote
4answers
55 views

$\mathbb{Z}_6/\mathbb{Z}_2$ isomorphic to $\mathbb{Z}_3$?

Recently in class my teacher mentioned that the quotient group $\mathbb{Z}_6/\mathbb{Z}_2$ is isomorphic to $\mathbb{Z}_3$. May I ask why is this so? Also, what do elements in ...
4
votes
1answer
15 views

Is a subring contained in the centralizer of its centralizer?

Fix a ring $B$. Given a subring $A \subset B$, we define$$A^! := \{b \in B : ab = ba,\text{ }\forall\,a \in A\},$$the centralizer of $A$ in $B$. This is a subring of $A$, so we can iterate $A^{!!} := ...
0
votes
1answer
22 views

If $G_1/N \unlhd G/N$ then $G_1 \unlhd G$?

I want to show that if $N$ is a simple normal subgroup of a group $G$ such that $G/N$ has a composition series, then also $G$ has a composition series. I think I can finish the proof if I can only ...
1
vote
1answer
25 views

Flatness via factoring homomorphisms

Theorem (4.32) of "Lectures on modules and rings" by T.Y. Lam says that a module $P_R$ is flat iff any $R$-homomorphism $λ:M→P$ where $M$ is any finitely presented $R$-module can be factord ...
1
vote
1answer
60 views

$(1)$ $a\boxplus b=a+b-ab$ for all $a,b \in F$. $(2)$ $a\boxdot b=1-t^{\log_t{(1-a)}\log_t{(1-b)}}$ for all $a,b\in F$

Let $1<t\in \mathbb{R}$ and let $F=\{a\in \mathbb{R}: a<1\}$. Define $\boxplus$ and $\boxdot$ on $F$ as follows: $a \boxplus b=a+b-ab$ for all $a,b \in F$. $a \boxdot b=1-t^{\log_t ...
0
votes
1answer
28 views

find all subgroups of G where: $0 \ne r \in \Re$ $G = <r>$

I need to find all subgroups of G where: $G \lt \Re$ $0 \ne r \in \Re$ $G = <r>$ $\Re$ is the group of real numbers and G is a subgroup. Edit : the operation is + I tried thinking about ...
5
votes
1answer
40 views

Given a group of order $p^nq^2$ for two odd primes, prove that the commutator is a p group.

Given a group of order $p^nq^2$ for two odd primes $p > q$, prove that the commutator is a p group. To solve this question I need to prove that the commutator can't be of the orders $p^iq$, ...
0
votes
0answers
15 views

proofing $Z(G)=\langle [x,u]\rangle$ if $M=C_G(u)$ is maximal subgroup

Let $G$ be non-abelian finite p-group, $p$ is odd, with cyclic center and $u\in G$ be of order $p$ if $M=C_G(u)$ (centralizer of $u$) be a maximal subgroup and $Z(G)\le M$ for $x\in G\setminus M$ how ...
0
votes
1answer
11 views

eigenvalues of cycle graph and its complement graph

I am trying to find the eigenvalue of cycle graph and its complement. How to simplify.Suppose $\omega^{1}+\omega^{n-1}=2cos (2\pi/n) $, then, $\omega^{\frac{n-1}{2}}+\omega^{\frac{n+1}{2}}=?$ Is it ...
6
votes
0answers
44 views

Explicit examples of (co)limit arguments in other fields

Over the past weeks, I have noticed that high level lecture notes in subjects like algebraic geometry, algebra, and algebraic topology often sketch proofs in the following form: Proof sketch ...
2
votes
0answers
24 views

calculation $p$-Fitting subgroup

Let $ G $ be a finite soluble group and $ A $ is the unique minimal normal subgroup of $ G $ that $ \vert A \vert = p^{a} $, $ p $ is prime. Let $ N =Fit(G) $, then $ N = O_{p}(G) $. Suppose $ F/N = ...
10
votes
4answers
1k views

Why is it necessary for a ring to have multiplicative identity?

I have read earlier that in a ring $(R,+,.)$ the following needs to hold: $(R,+)$ is an abelian group multiplication is associative and closed left and right distribution laws hold. However, I ...
2
votes
0answers
46 views

A finite von Neumann regular ring is unital and has $ab = 1$ if $ba = 1$

Let $R$ be a finite ring satisfying for any $x \in R$ there exists $y \in R$ with $xyx = x$. Show that $R$ is unital and that if $ab = 1$, then $ba = 1$. Thoughts so far: If I can show that the ...
1
vote
1answer
39 views

$x$ and $g$ are elements of the group $G$, show that the order of $x$ is equal to the order of $g^{-1} xg$.

If $x$ and $g$ are elements of the group $G$, prove that $|x| = | g^{-1} xg|$. Deduce that $|ab| = |ba|$ for all $a,b \in G$. attempt: Let $|x| = n$ be the order of $x$ and $| g^{-1} xg| = m$ be the ...
2
votes
2answers
44 views

Existence of integer solution of $a^2 -17b^2 = $ any constant

When checking whether if $9-\sqrt{17}$ in the ring $\{a+b\sqrt17: a,b \in \mathbb{Z}\}$ is a prime. Suppose $$\alpha\cdot \beta = 9-\sqrt{17},$$ using norm argument $$N(\alpha)N(\beta) = ...
3
votes
1answer
54 views

Is there a nice expression for $f(x) = (1+x)(1+x^2)(1+x^3)\cdots$

While I was solving a problem, I stumbled upon this function $$f(x) = (1+x)(1+x^2)(1+x^3)\cdots$$ I tried to write out the first few products but I couldn't recognize any meaningful pattern. Is there ...
0
votes
1answer
33 views

Is a finite monoid with left cancellation property always a group?

I need to answer and show if a Monoid with left cancellation property always a group. I managed to show that it is correct when cancellation property holds for both left and right (that was part a of ...
-1
votes
2answers
28 views

Understanding the connection of the roots of an irreducible polynomial and a basis for field extensions

Let $\alpha,\beta,\gamma \in E$ be the roots of an IRREDUCIBLE polynomial $p(x)\in Q[x]$ (where E/Q is an extension field. Can I use these roots to construct a basis for E over Q? Why?
5
votes
0answers
38 views

For finitely generated free abelian groups $A,B$ if there is an onto homomorphism $A \to B$, then $\operatorname{rank}(A) \geq \operatorname{rank}(B)$

$\newcommand{\rank}{\operatorname{rank}}$For two finitely generated, free abelian groups $A,B$ prove that if there is an onto homomorphism $A \rightarrow B$, then $\rank(A) \geq \rank(B)$ Assume that ...
2
votes
1answer
27 views

How to create a ring in MAGMA with relations?

I'm using MAGMA221 and would like to create a ring over $GF(2)$ with respect to a list of relations. Here's what I have so far: $\mathtt{Z:=GF(2);} \\\mathtt{P<x,y,z>:=PolynomialRing(Z,3);}$ ...
0
votes
0answers
28 views

Prove that the center of a group with $385$ elements has an element of order 7. [duplicate]

Prove that the center of a group with $385$ elements has an element of order 7. By Cauchy's theorem I know that if I prove that $Z(G)$'s order is divisable by 7, we're done. So now I need to rule out ...
1
vote
0answers
30 views

When the fibers of $A \to A[T]/(h(T))$ are geometrically regular?

Let $A$ be an affine commutative noetherian domain over a characteristic zero field $K$, $T$ an indeterminate, $h=h(T) \in A[T]$ (not necessarily monic), $B=A[T]/(h)$, and assume that $(h)$ is a prime ...
-1
votes
1answer
34 views

Empty set - group [duplicate]

I started a course of algebra this morning, and the teacher explained the structure of a group. He explicitly explained a group has to be empty. Someone can explain to me why it is a necessity for a ...
0
votes
2answers
53 views

Between complex numbers and quaternions?

Complex numbers are $a+ b i $; Quaternions are $a + b i + c j + d k $. So, do there exist numbers like $a + b i + c j$? Here $a$, $b$, $c$, $d$ are all real. Did Hamilton consider such a case?
0
votes
1answer
40 views

Let $V=\{i\in \mathbb{Z}: 0\leq i< 2^n\}$. Define vector addition and scalar multiplication on $V$ to turn it into a vector space over $GF(2)$.

Let $V=\{i\in \mathbb{Z}: 0\leq i< 2^n\}$ for some $n\in \mathbb{N}$. Define vector addition and scalar multiplication on $V$ in such a way as to turn it into a vector space over the field ...
2
votes
1answer
24 views

Evolution operator always in $SU(n)$?

Think about the evolution operator $U$ in Quantum Mechanics for finite dimensional systems. Then this operator satisfies an equation $$U'(t) = -iHU(t)..$$ Here, I assume that $H$ is ...
1
vote
1answer
39 views

How to prove that O(Ng) | O(g)

I have this exercise: Let $G$ be a group and $N$ a normal subgroup of $G$. Show that for all $Ng\in G/N$, $$o(Ng)\mid o(g).$$ For now, without using the canonic homomorphism $\tau ...
2
votes
2answers
164 views

Group Action as permutations

I'm trying to study on Group actions. the paper says(if I understand) that if I have Set $S$ and an action $\alpha$: $G \times S \rightarrow S$ . then the action may be viewed as permutation by $x ...
0
votes
1answer
36 views

Are subgroups of order $p^{n-1}$ maximal?

Let $G$ be finite p-group of order $p^n$, I know all maximal subgroups of order $p^{n-1}$ Is it right to say all subgroup of order $p^{n-1}$ are maximal subgroups? If not, what property $G$ must have ...
0
votes
0answers
23 views

Rings leading to AKS primality test

Given number n, define ring $R = \Bbb Z_n[x]/(x^r −1)$ for a carefully chosen number $r$ ($r$ is much smaller than $n$; of the order of square of the number of digits in $n$) An element of $R$ is a ...
0
votes
2answers
47 views

Group of an order 385

Let $G$ be a group of order $385$, proof that $Z(G)~~ (cent(g))$ contains object of order $7$. I used sylow theorm and realized that there are $1$ sylow-$11$ sub-group which is normal in $G$ and also ...
1
vote
0answers
18 views

Enumerating functions modulo action on both the domain and codomain.

Let $Hom(A,B)$ be the set of functions from a finite set $A$ to a finite set $B$ and let $G_A \leq S_A$, $G_B \leq S_B$ be a subgroups of the permutation groups of $A$ and $B$. For $f,g \in Hom(A,B)$ ...
1
vote
0answers
28 views

Factor rings $R/R$ and $R/0$

Let $R$ be a ring. I want to describe the factor rings $R/R$ and $R/0$. So $R/R = \{[r]| r+R, \forall r\in R \}$ and since $r+R=R$, we get $R/ R =\{[0]\}$. And for $R/0 = \{[r]| r+0,\forall r\in ...