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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Group presentations and subgroups

How to prove that $G=\langle a,b,c\mid a^2 = b^2 = c^3 = 1, ab = ba, cac^{-1} = b, cbc^{-1} =ab\rangle$ has no subgroup of order $6$ without finding $G$? $\bf Edit$: Given that $|G|=12$.
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Subgroup complement for normal subgroup of $G$ with trivial center and $\mathrm{Aut}(N)=\mathrm{Inn}(N)$

Let $G$ be a finite group and $N \subseteq G$ be a normal subgroup. If $Z(N)$ is trivial and $\operatorname{Aut} N=\operatorname{Inn} N$, then show $N$ has a complement $H$ and $H$ is normal in $G$ ...
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163 views

Algebraic structure of a set of Egyptian fractions of a positive rational?

It is said that every positive rational number can be represented by infinitely many Egyptian fractions (defined as the sum of distinct unit fractions). I am struggling to understand in a formal way, ...
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241 views

A presentation of a group of order 12

Show that the presentation $G=\langle a,b,c\mid a^2 = b^2 = c^3 = 1, ab = ba, cac^{-1} = b, cbc^{-1} =ab\rangle$ defines a group of order $12$. I tried to let $d=ab\Rightarrow G=\langle d,c\mid ...
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How to make existence proof for Abelian Group condition 3( unit element e), when ($\mathbb{N}. \cdot)$?

How to make existence proof for Abelian Group condition 3( unit element e), when ($\mathbb{N}. \cdot )$, where $\cdot$ is natural multiplication? Is it by example? Is it done by constructive proof ...
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1answer
55 views

upper central series, taking intersection with subgroup ($\zeta_i G \cap H \leq \zeta_i H$)

Let $\zeta_i G$ be the upper central series of $G$. Show that for 'any' subgroup $H$, we have $$\zeta_i G \cap H \leq \zeta_i H$$ where $\zeta_i H$ is the upper central series of $H$. I tried ...
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2answers
127 views

Prove: $f(x)^{p^k}\equiv f\left(x^{p^k}\right)\bmod p$

$p$ is a prime number, $k$ is an positive integer, and $f\in\Bbb Z[x]$. Prove: $f(x)^{p^k}\equiv f\left(x^{p^k}\right)\bmod p$
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49 views

question on group representations

Here is a problem I faced in algebra. $\rho: A_4 \rightarrow End_{\mathbb C}\mathbb C^{10}$ is a representation of $A_4$. Then show that there is a vector $v \in \mathbb C^{10}$ such that $v$ is an ...
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1answer
105 views

To What Extent Does the Cartesian Product for Algebraic Structures Generalize?

I admit this question is quite general. If we have a group (or perhaps some other algebraic structure) $G$, we can define the Cartesian product $G\times G$ of $G$ with itself. And then powers of $G$ ...
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2answers
441 views

A prime ideal $I$ in $\mathbb Z[\sqrt{-5}]$ so that $7\in I$.

Find a prime ideal $I$ in $\mathbb Z[\sqrt{-5}]$ such that $7\in I$. I claimed that $I= 7\mathbb Z[\sqrt{-5}]$, and tried to prove that if $x,y\in \mathbb Z[\sqrt{-5}]$ and so that $xy\in I$ then ...
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152 views

Let $F= \Bbb{Q}(\zeta_9)$ with $\zeta_9 = e^{\frac{2i\pi}{9}}$. Find the Galois group and the intermediate fields.

Let $F= \Bbb{Q}(\zeta_9)$ with $\zeta_9 = e^{\frac{2i\pi}{9}}$. a) What is the Galois group of $F$ over $\Bbb{Q}$? b) Find all intermediate fields between $\Bbb{Q}$ and $F$. (Write each in ...
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1answer
119 views

Check existence of an isomorphism in four examples

Check if there exists an isomorphism for : $$ i) \ G = (\mathbb{Z}_{15}, +_{15}), \ H = (\mathbb{Z}_{15}, +_{15}), \ f(1) = 2 $$ $$ ii) \ G = (\mathbb{Z}_{15}, +_{15}), \ H = (\mathbb{Z}_{15}, ...
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3answers
146 views

Frieze groups, wallpaper groups

Can someone suggest a source that proves the classifications of the 7 frieze groups and 17 wallpaper groups in an elegant way?
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1answer
646 views

A submodule of a free module is torsion-free?

I am studying for a comprehensive exam and looking at a large bank of problems. One problem has six statements about module and asks for a a proof or a counter-example of the statements. I am able ...
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109 views

Quotient field of a certain quotient ring

Let $A$ be a commutative integral domain and $\mathfrak p$ a prime ideal of $A$. Let $A_{\mathfrak p}$ be the localization of $A$ at $\mathfrak p$ and $\mathfrak{m}_{\mathfrak{p}}=\mathfrak ...
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242 views

On composition of polynomials

Given two irreducible polynomials $f_{u}(x),f_{r}(x) \in \Bbb Q[x]$, can one find two polynomials or rational functions $h_{u}(x),h_{r}(x) \in \Bbb Q[x]$ or $\Bbb Q(x)$ respectively such ...
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3answers
97 views

$G$ a group of odd order. Then $\forall$ $g\in G$ there is $h\in G$ such that $g=h^2$

This one is from a practice exam I was working on. $G$ a group of odd order. Then for $\forall$ $g\in G$ there is a unique $h\in G$ such that $g=h^2$. Thoughts Well I tried a few things but they ...
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2answers
142 views

Check condition normal subgroup in these three examples

Is the subgroup H of G is a normal subgroup of G, for: $$ i)\ G = S_5, \ H = \{id, (1,2)\} $$ $$ ii) \ G = (Sym(\mathbb{N}), \circ), \ H = \{f\in Sym(\mathbb{N}) : f(0) = 0 \}$$ $$ iii) \ G = S_4, \ H ...
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2answers
628 views

Free modules over commutative rings. [duplicate]

Free modules over a commutative ring $R$ with $1$ have well-defined rank. I have been wondering if there is a ring $R$ such that there are free modules $M'\subset M$ with ...
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0answers
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Commutativity of s-unital rings.

Theorem. Let $R$ be a left (resp. right) s-unital ring. If $R$ satisfy $(P_1)$ (resp. $(P_2)$). Then R is commutative(and conversely). $(P_1)$ $y^{s}[x,\, y]=\pm x^{p}[x^{m},y^{n}]^{r}y^{q}$ where ...
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957 views

Torsion Subgroup (Just a set) for an abelian (non abelian) group.

Let $G$ be an abelian Group. Question is to prove that $T(G)=\{g\in G : |g|<\infty \}$ is a subgroup of G. I tried in following way: let $g_1,g_2\in T(G)$ say, $|g_1|=n_1$ and $|g_2|=n_2$; Now, ...
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115 views

Let $G$ be finite abelian group. Determine $n$ such that following sentence is true.

Determine $n \in \mathbb{N}$ such that following sentence is true: Exists finite abelian group and exists $a,b \in G$ such that $\mbox{ord}(a)=6, \mbox{ord}(b)=10, \mbox{ord}(ab)=n$. I suppose that ...
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2answers
360 views

Prove that if for all $aba=bab$ then $|G|=1$.

Let $G$ be a group such that for all $a,b \in G$ we have $aba=bab$. Prove that $|G|=1$. So I have to show that $G =\left\{e \right\} $. Because for any $a,b \in G$ we have $aba=bab$, let $b=e$. Then ...
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1answer
36 views

is it a group homomorphism?$f:(\mathbb{R}^*,\cdot)\to(\{-1,+1,\},\cdot)$ defined by $f(x)={x\over |x|}$

$f:(\mathbb{R}^*,\cdot)\to(\{-1,+1,\},\cdot)$ defined by $f(x)={x\over |x|}$ , is it a group homomorphism? does it make sense?I am confused to see the question in a past year question paper. I know ...
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11answers
2k views

What if $\pi$ was an algebraic number? (significance of algebraic numbers)

To be honest, I never really understood the importance of algebraic numbers. If we lived in an universe where $\pi$ was algebraic, would there be a palpable difference between that universe and ours? ...
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131 views

Simple question about cyclic submodule

Let $R$ be a ring, $A$ an $R$-module, $a \in A$. Then the cyclic submodule $C$ generated by $a$ is $\{ra +na \mid r\in R,n\in \mathbb{Z}\}$. Here I am using the definition of a cyclic (sub)module ...
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440 views

Cardinality of the quotient ring $\mathbb{Z}[x]/(x^2-3,2x+4)$

This problem is from a practice exam I was working on. What is the cardinality of the quotient $\mathbb{Z}[x]/(x^2-3,2x+4)$ ? Thoughts. If I find a ring that is easier to handle then this then I ...
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2answers
220 views

Let $G$ be a group of order $4p^n$, where $p>2$ is prime and $n>0$. Show that $G$ is not simple.

Let $G$ be a group of order $4p^n$, where $p>2$ is prime and $n>0$. Show that $G$ is not simple. (Hint: Consider the standard action of $G$ on $G/P$, where $P$ is a $p$-sylow subgroup.) Let ...
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0answers
85 views

Show that a periodic, finitely generated, nilpotent group has finite order.

Show that a periodic (every element has finite order), finitely generated, nilpotent group has finite order.
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133 views

A non-Archimedean norm definition can be strengthened.

See Andrew Baker's p-adic notes: For a non-Archimedean norm $N$ it is true that "$N(x + y) \leq \max\{N(x), N(y)\}$, with equality if $N(x) \neq N(y)$." Having trouble proving this. Please ...
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0answers
63 views

Minimal prime ideals are made of zero-divisors [duplicate]

Let $R$ be a commutative ring with unity which is not an integral domain. Let $P$ be any minimal prime ideal of $R$. How can I show that $P⊆Z(R)$, where $Z(R)$ denotes the set of zero-divisors of $R.$ ...
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Trying to prove structure result for ${\rm Hom}(A,B)$

Let $A$ be a $\mathbb Z$-algebra that is finitely-generated and free as a $\mathbb Z$-module and let $\pi: A \rightarrow \mathbb Z$ a nontrivial $\mathbb Z$-module homomorphism. For a positive ...
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165 views

Find all homomorphisms in these three examples

How I should find all homomorphisms $$ f : G \rightarrow H $$ for examples: $$ i) \ \ G = (Z, +), \ \ H = (Q, +) $$ $$ ii) \ \ G = (Z_{15}, +_{15}), \ \ H = (Z_4, +_4) $$ $$ iii) \ \ G = Z_2 \times \ ...
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3answers
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Prove that if $x^2=e$ then order of $x$ is $1$ or $2$

Let $(G,*)$ be a finite group and $x$ an element of $G$. Prove that if $x^2=e$ then the order of $x$ can be only $1$ or $2$?
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Irreducible Elements, Units, UFD

Let $P$ be a set of positive prime numbers. Let $\mathbb{Z}_{P}$ be the collection of all rational numbers of the form $a/b$, where $a,b$ are integers, $b$ not in $0$, and for all $p \in P$, $p$ ...
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5answers
262 views

Why composition in $S_3$ is not commutative

The family of all the permutations of a set $X$, denoted by $S_X$, is called the symmetric group on $X$. When $X = \{1, 2, \dots , n\}$, $S_X$ is usually denoted by $S_n$, and it is called the ...
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A group of order 16 has a normal subgroup of order 4

Let $ G$ be a group of order $16$. Show that $G$ must contain a normal subgroup $H$ of order $4$. I tried the Sylow first theorem, that is $\{e\}\triangleleft H_1\triangleleft H_2\triangleleft ...
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Pfister's 16-Square Identity and the norm of sedenions

Consider the sequence of numbers: complex numbers $\Bbb C$, quaternions $\Bbb H$, octonions $\Bbb O$, and sedenions $\Bbb S$. The Brahmagupta-Fibonacci 2-Square identity implies that the norm of the ...
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Counter examples in group theory

Let $G$ be a finite group with normal subgroups $N_{1}$ and $N_{2}$. Find counter examples to the following statements 1) If $N_{1}\cong N_{2}$ then $G/N_{1}\cong G/N_{2}$ 2) If $G/N_{1}\cong ...
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On a PID that is not an Euclidean domain

Let $\omega = \frac{1 + \sqrt{19}i}{2}$. The article here claims to prove that $\mathbb{Z}[\omega]$ is an example of a PID which is not an Euclidean domain. To prove that it is a PID, it takes an ...
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How the ring of algebraic numbers looks like?

Suppose I have an algebraic number field $K = \mathbb Q(\alpha)$ for some $\alpha \in O_K$, ring of algebraic integers. Is there a criterion that tells us when $O_K =\mathbb Z[\alpha]$ by any ...
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72 views

An example of s-unital rings

I am just studying s-unital rings. A ring is called left (resp. right) s-unital if $x\in Rx$ (resp. $x\in xR$) for all $x$ in $R$. A ring is called s-unital if and only if $x\in xR\cap Rx$ for all ...
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216 views

Non-abelian groups where $(ab)^{n+i}=a^{n+i}b^{n+i}$ for all $0\leq i\leq k$, $k>1$

In this question it is asked for an example of a group $G$ such that $(ab)^n=a^nb^n$ for all $a, b\in G$ holds for two consecutive integers $n\in\{m, m+1\}$, but $G$ is not an abelian group. The ...
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76 views

An irreducible polynomial of degree $4$ in $\mathbb{Z}_5[x]$

Q: Find an irreducible polynomial of degree four in $\mathbb{Z}_5[x]$. My Answer: $x^{4} - 2$ Is my answer correct?
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Rank of free modules under reduction

Suppose $R$ is a principal ideal domain, $M$ is a free $R$-module of rank $n$ and $f$ is any $R$-module homomorphism from $M$ to $R$. If $\mathfrak a$ is a non-trivial ideal in $R$ and ...
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193 views

If $x,y,z \in \mathbb{Z}$ such that $x^4+y^4+z^4 \equiv 0 \pmod{29}$, prove that $x^4+y^4+z^4 \equiv 0 \pmod{29^4}$

If $x,y,z \in \mathbb{Z}$ such that $x^4+y^4+z^4 \equiv 0 \pmod{29}$, prove that $x^4+y^4+z^4 \equiv 0 \pmod{29^4}$. I have no idea where to start, but this is my abstract algebra homework, so I ...
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3answers
186 views

Please list a few topological groups that I should learn about.

I'm going through Munkres' Topology book and there's a lot about topological groups. For fear that I'll forget the theorems on them I'd like to connect each thing I prove with a real-world example. ...
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1answer
86 views

Find a polynomial ring $R$ which is not an integral domain and an ideal $I$ such that $R/I$ is a field.

Question: Find a polynomial ring $R$ which is not an integral domain and an ideal $I$ such that $R/I$ is a field. Answer: $R=\mathbb{Z}_6[x]$, $I=\langle 2,x\rangle$, $R/I$ is isomorphic to ...
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1answer
94 views

Minimal Polynomial Trouble

Let $L$ be an extension of a field $F$. Let $\alpha_1, \alpha_2\in L$ be such that both of them are algebraic over $F$ and have the same minimal polynomial $m$ over $F$. We know that there is an ...
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135 views

Uniqueness of the endomorphism of the multiplicative group of positive real numbers

How do we prove that the endomorphism of the multiplicative group of positive real numbers is unique (up to a complex variable)!? meaning: how do we prove that it has the following - and only the ...