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.

learn more… | top users | synonyms (1)

3
votes
2answers
39 views

Group having an element $x$ of order $p$ where $p$ is the smallest prime dividing |G|

Let $G$ be a finite group and $p$ be the smallest prime dividing $|G|$ and $x\in G$ be an element of order $p$. Let $h\in G,$ and $hxh^{-1}=x^{10}$. Then prove that $p=3$. If $H=<h>, ...
0
votes
1answer
16 views

Part of a proof in Herstein about Gaussian Integers being a Euclidean Ring

In Herstein topics in algebra (2nd Edition) page 150, in proof of theorem 3.8.1, in the first special case where $n$ is a positive integer and $y=a+bi$ is a general Gaussian integer, where he is ...
0
votes
1answer
12 views

Fields with Additive identity powers

Would it be possible to have a field (or field-like structure) with an additive identity $k$ where $k^a\neq k^b$ for $a\neq b$? I need this because I'm working with a field-like structure where if I ...
0
votes
2answers
50 views

Is there any motivation for the direct sum?

I believe that everything have reason to exist . I want to learn why the direct sum exist. Is there any motivation for the direct sum to exist ?
1
vote
0answers
25 views

Abstract algebra notation

I am new to learning abstract algebra and using multiple books but the notation varies enough to throw me off. Could someone explain to me the differences between: $\mathbb{Z}\left\{ p\right\}$ ...
1
vote
1answer
21 views

Question about proof about index and subgroups

Let $G$ be a group so that $H\lhd G$. There is an element $g \in G$ so that $g$ isn't in $H$ but $g^2$ is in $H$. Show the index is even. Can't I just say that the cosets of $H$ are $H$ and $Hg$ ...
1
vote
3answers
25 views

A question about normal subgroups and index

Let $G$ be a group, and $H$ be a normal subgroup of $G$. $|H|=11$ and $[G:H]=24$. Let there be $x \in G$ and $x^{11}=e$. Show $x \in H$. Would like hints etc' on how to solve this. Is proving that ...
1
vote
1answer
31 views

Groups with single conjugacy class of subgroups. (modified)

I am going to modify my previous question. What are those finite non abelian groups in which non normal subgroups of same order are conjugate. e.g. Dihedral groups of order $4n+2$.
3
votes
3answers
55 views

Why is the reciprocal of an $n$-th root of unity its complex conjugate?

As stated in the Wikipedia article on roots of unity, the reciprocal of an $n$-th root of unity is its complex conjugate. They provide the following proof of this statement: Let $z\in\mathbb{C}$ be a ...
4
votes
6answers
63 views

Why is $\mathbb{R}/\mathbb{Z}$ isomorphic to the complex numbers of length one?

Wikipedia states that the quotient group $\mathbb{R}/\mathbb{Z}$ is isomorphic to all complex numbers of length $1$. I have a hard time making sense out of this, and in particular, how complex numbers ...
1
vote
1answer
33 views

Let $G$ be a group. Then verify the statements with justification:

Let $G$ be a group. Then verify the statements with justification: $\bullet$ If $G$ has nontrivial centre $C$, then $G/C$ has trivial centre. $\bullet$ If $G$ does not equal $1$, there exists a ...
0
votes
0answers
25 views

Describing $k$-automorphisms

(1) Let $B=k[T]$ with $k$ a field. Describe the group $Aut_kB$ of $k$-automorphisms of B. (2) Do the same for the field $K=k(T)$ or rational functions.
0
votes
1answer
43 views

Question about group theory and order in $\mathbb Z_n$

This is a only theoritical. Why is the order $o( \bar x )$ of $\bar x∈\mathbb Z_n$ the smallest non-negative integer $k$ such that $kx \equiv 0$ (mod $n$)? I don't understand how it follows from ...
0
votes
0answers
14 views

How to show that a cyclic group is partitioned into these classes under a equivalence relation.

Condition: $S=$ {$h^k|k\in \mathbb Z$}, $h^{k_1}\sim h^{k_2}$ iff $k_1 \equiv k_2$ $mod$ $n$, Then, how to show that $S=[h^0]\bigcap[h^1]\bigcap[h^3]\bigcap...\bigcap[h^{n-1}]$?
3
votes
3answers
62 views

Only $1$ Nontrivial Subgroup $\Longrightarrow |G| = p^2$ [duplicate]

I am pretty new to this site , so I am not sure how things work, but I am in desperate help with a question that I don't know where to start or finish with. It is for a test I have to study for. Here ...
1
vote
1answer
38 views

Can we give of the fact that a group of order $9$ is abelian without using an argument involving the product of two cyclic groups of order $3$?

A group of order $9$ is always abelian. I've seen proofs of this result, but I would like to prove it the following way: Let $G$ be a group of order $9$. If $G$ has an element $a$ of order $9$, then ...
0
votes
3answers
37 views

Show that $(3,1)$, $(-2,-1)$, and $(4,3)$ generate the additive group $\mathbb{Z} \times \mathbb{Z}$

Show that $(3,1)$, $(-2,-1)$, and $(4,3)$ generate the additive group $\mathbb{Z} \times \mathbb{Z}$. I need your help.
2
votes
1answer
42 views

Visualizing the 48 actions of GL(2,3)

Hello and thank you for your patience. (DISCLAIMER: I'm a novice and not a mathematician by trade and I'm not certain how to articulate most of my questions here. I am learning from experiences and ...
-2
votes
1answer
27 views

Generators of the additive cyclic group $\mathbb{Z}$

Show that the only generators of the additive cyclic group $\mathbb{Z}$ are $1$ and $-1$. I need your help.
2
votes
1answer
50 views

What sort of algebra is this?

Let us say that I have a set of symbols, $S$. The symbols can be operated on by a set of $n$-ary operators, $O$. Importantly, some of these operators are in the set of symbols, i.e. $S \cap O \neq ...
2
votes
2answers
53 views

What is the most elementary proof of the following: a non-abelian group of order $6$ is isomorphic to $S_3$

We know that, a non-abelian group of order $6$ is isomorphic to $S_3$. While I've been able to locate different proofs of this result, I would like to have one that is as elementary as it can be. So ...
1
vote
1answer
39 views

An assumption used in defining a group of order $mn$

I want to show that the defining relations $a^m=b^n=e, ab=ba^k$ define a group of order $mn$ with a normal subgroup of order $m$, if $k^n \equiv 1 \pmod m$. Consider the set of symbols of the form ...
3
votes
1answer
47 views

How do I effectively solve this kinds of problems?

I'm preparing for a test on Monday. This is an exercise in the past homework. Find all subgroups of $\mathbb{Z}_2\times \mathbb{Z}_2\times\mathbb{Z}_4$ isomorphic to $Z_2\times Z_2$. Is there a ...
0
votes
1answer
25 views

Lie bracket of vector fields on $R^2$

Compute the Lie bracket$$\Big[-y\frac{\partial}{\partial x}+x\frac{\partial}{\partial y},\frac{\partial}{\partial x}\Big]$$ on $R^2$ Can you help me please?
1
vote
1answer
35 views

The first Weyl algebra is Calabi-Yau

Why is the Weyl algebra $A_1(k)$ over a field $k$ Calabi-Yau? (My definition of Calabi-Yau is Ginzburg's)
3
votes
3answers
84 views
+50

Non simplicity of a group of order $p^{100}q$ given some conditions.

Let $G$ be a group $p$, $q$ primes $p\neq q$, $|G|=p^{100}q$. 1.- Assume that $G$ has two different $p$-Sylow subgroups and intersection for each pair of $p$-Sylow subgroups is trivial. Show that ...
0
votes
1answer
22 views

Why the rank of finitely generated finite abelian group is 0?

I haven't proven 'the fundamental theorem of finitely generated abelian group' Nevertheless, it's written in my text(Dummit,Foote - p.159) it's gonna be proven in a later chapter. Also, it's written ...
1
vote
2answers
41 views

$\mathbb{Z}/(p)^n$ does not contain a field if $n \geq 2$?

Show that $\mathbb{Z}/(p)^n$ does not contain a field if $n \geq 2$ and cannot be made into a vector space over $\mathbb{F}_p$ in a compatible way with its Abelian structure.
6
votes
2answers
93 views

Minimal polynomial: is $\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}=1$?

I was wondering about the minimal polynomial of real number $$u=\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}$$ over field $\mathbb{Q}$. As you can see here, I worked out that $u$ is a root of monic ...
0
votes
1answer
15 views

what is the definition of linearly independent subset of abelian group?

WHat is the definition of linearly independent subset of abelian group? I know the concept of this, but i don't know how to define this term precisely. Below is what i tried to formulate: Let ...
2
votes
1answer
60 views

Which non-Abelian finite groups contain the two specific centralizers? - part II

This is a question requiring the good knowledge of group theory: (Q1) Which finite groups $G$ contains some specific centralizers isomorphic to both of these two groups (but may contain other ...
2
votes
0answers
50 views

Which finite groups contain the two specific centralizers?

This is a question requiring the good knowledge of group theory: (Q1) Which finite groups $G$ contains some specific centralizers both of these two groups: i. the elementary group $Z_2^4$, ...
0
votes
2answers
57 views

Subgroup of group of order $44$

Pick the correct statement(s) below: $(a)$ There exists a group of order $44$ with a subgroup isomorphic to $ Z_2 + Z_2 $. $(b)$ There exists a group of order $44$ with a subgroup isomorphic to $ ...
0
votes
0answers
28 views

To Find a subgroup of order $6$ in $U(700).$

Find a subgroup of order $6$ in $U(700).$ Attempt: $U(700)=U(2^2.5^2.7)\thickapprox U(2^2) \oplus U(5^2)\oplus U(7)$ $\thickapprox \mathbb Z_2 \oplus \mathbb Z_{5^2-5} \oplus \mathbb Z_{7-7^0}$ ...
2
votes
0answers
19 views

normal group $(\text{ker }g)(\text{ker } f)$ as a kernel of some group using $f$ and $g$

For group homomorphism $f: A\to B$ and $g: A\to C$ we know $\text{ker }f\cap \text{ker } g$ is kernel of $(f,g): A\to B\times C$. $(\text{ker }g)(\text{ker } f)$ is trivially normal in $A$, can we ...
0
votes
0answers
31 views

Showing that a set is Discrete Valuation Ring

An order function on a field $K$ is a function $ϕ:K→Z∪{∞}$ satisfying: i) $ϕ(a)=∞$ if and only if $a=0$. ii) $ϕ(ab)=ϕ(a)+ϕ(b)$. iii) $ϕ(a+b)≥min(ϕ(a),ϕ(b))$. Show that R=$\{z∈K∣ϕ(z)≥0\}$ is a DVR ...
2
votes
1answer
60 views

Subfield of the Galois Group of $x^5 - 1$

Why is it that the subfield fixed by the subgroup of this Galois group is $\mathbb{Q}(\sqrt5)$. Can someone explain it without using the cyclotomic extension of $\mathbb{Q}$? Thank you edit: Using ...
0
votes
1answer
18 views

Separable and Splitting fields

Does a separable field imply splitting field? I am thinking yes. For $F < E$ If every $\alpha \in E$ is separable over $F$ then it must be that $E$ is a $S.F.$ over $F$? Also we have the ...
2
votes
2answers
39 views

Sub-modules of free modules

I'm going back through basic module theory notes, and I've come across a paragraph explaining that a sub-module of a finitely generated free module may not itself be free. In my course a free module ...
0
votes
1answer
61 views

Ring of fractions $S^{-1}A$ and localisation

I'd really appreciate if somebody could help me with the problem 6.4 Reid (Undergraduate commutative algebra), because I've been trying to get the solutions for days and I don't see it. (a) Give an ...
3
votes
0answers
40 views

If $A \in M_{n,n}(\mathbb F)$ is invertible then $A = UPB$, $U$ is unipotent upper triangular, $B$ is upper triangular and $P$ is a permutation.

If $A \in M_{n,n}(\mathbb F)$ is invertible then $A = UPB$, where $U$ is unipotent upper triangular, $B$ is upper triangular and $P$ a permutation matrix. A hint is given that one could relate ...
1
vote
1answer
51 views

Is this type of a subgroup always normal? [duplicate]

Let $G$ be a finite group, and let $H$ be a subgroup of $G$ such that the index $(G\colon H)$ of $H$ in $G$ is the smallest prime that divides the order of $G$. Can we say anything about whether or ...
1
vote
1answer
24 views

Obtaining representations of $G$ from $\mathrm{Lie}(G)$.

Suppose $\mathfrak{g}$ is a semisimple Lie algebra over $\mathbb{C}$, and $\tilde{G}$ is the unique connected, simply connected Lie group whose Lie algebra is $\mathfrak{g}$. Let $C$ be any discrete ...
0
votes
1answer
27 views

When proving a partial order relation is a total order do we have assume both elements are distinct?

Consider the "divides" relation on the set $A=\lbrace 1,2,2^2,.\;.\;.,2^n\rbrace$, where $n$ is a non-negative integer. Prove that this relation is a total order on $A$. First we prove $A$ is a ...
1
vote
1answer
64 views

Need an explanation for homomorphism in commutative algebra

I'm self-learning commutative algebra following "Introduction to Commutative Algebra". When dealing with concepts like "contraction" and "extension", some exercises in this book don't specify which ...
0
votes
1answer
40 views

If G/Z(G) is abelian, then H/Z(H) is also abelian.

Let H be a subgroup of G. If G/Z(G) is abelian, then H/Z(H) is also abelian. I am trying to prove this property yet I don't know how I should proceed. Any help? Thank you.
0
votes
0answers
27 views

Proof about dual bases?

Let V be a finite dimensional vector space over a field F. Let B={v1,v2, ..., vn} be a basis and consider the dual basis B*={v1*,v2*,...,vn*}. Let a be an element of V*. prove that $$v = ...
0
votes
1answer
24 views

Do disjoint cycles commute?

When a given set is finite it is clear. I'm asking the general case. Let $X$ is an arbitrary set. Let $\sigma,\tau$ be disjoint cycles on $X$. Then do they commute?
0
votes
1answer
25 views

Direct Products Help Abstract

Let Z be the additive group of integers and S = {-1,1} be a group under multiplication. Is the product Z x S cyclic? Why or why not? I am really confused on this question and have no idea where to ...
0
votes
1answer
21 views

Abstract Direct Product Proof Help

Let G = G1 x G2. Let H = {(x1, e2) : x1 ∈ G1} and K = {(e1, x2) : x2 ∈ G2}. (a) Prove H ≤ G and K ≤ G. (b) Prove that HK = KH = G (c) Prove that H ∩ K = {(e1, e2)} (d) Show that G/H is isomorphic to ...