0
votes
1answer
25 views

Greatest common divisor of ring elements

Consider the ring $\mathbb Q[x]$. (a) Suppose that $a(x) = (x+1)^3(x-1)^4(x+2)$ and $b(x) = (x+1)^2(x+2)^3(x-3)^4$. What is the $\gcd (a(x),b(x))$? (b) Suppose that $c(x) = (x^2-1)^4(x^2+3x+2)$. ...
1
vote
2answers
33 views

Ring with special rules for add and mult

$R$ is a ring of integers with special rules for multiplication and addition. Suppose that $f: \mathbb Z \to R$ is an isomorphism that is defined by $f(x) = x-2$. What integer in the ring $R$ is ...
0
votes
1answer
25 views

GCD of polynomials

In $\mathbb{Z}/5\mathbb{Z}[x]$ use the Euclidean Algorithm to find the GCD of $x^4+x^2$ and $x^4+x^3+3x$. My thoughts: I am getting to the point where I need to use a fraction to get further in ...
0
votes
0answers
15 views

Question regarding a surjective function

The function $f :\mathbb{Z}_{12} \longrightarrow \mathbb{Z}_4$ is defined by $f(x)=3x$. I am asked given any $a,b \in \mathbb{Z}_{12}$ if preservation of addition and multiplication occur. I proved ...
1
vote
1answer
34 views

Splitting field in multiple field extension

Let $F\subset K\subset E$ be fields, $f(x)\in F[x]$. E is the splitting field of $f(x)$ over $F$. Prove $E$ is the splitting field of $f(x)$ over $K$. My problem with this proof is that I do not know ...
0
votes
1answer
22 views

Proving two cosets are either equal or disjoint

I have a proof that I must produce, but I'm a little unsure of how to structure it, as I don't have a great deal of practice in forming rigorous proofs. We have, G the group of 2x2 invertible ...
3
votes
1answer
68 views

Rings (integral domain and fields)

True or false: (1) Every integral domain is a field (2) every field is an integral domain (3) the ring $\mathbb Z$ is a field. (4) the ring $\mathbb Z/(17)$ is a field. (5)The set $\{[0], [2], ...
1
vote
2answers
33 views

Associates in Domains

Let D be a domain and $a, b \in D^*$. Show that $a$ is a proper divisor of $b$ if and only if $b=ax$ for some nonzero nonunit $x$. I'm just really not sure how to start this. Any advice would be ...
0
votes
1answer
32 views

Proving the set of the $n$-th roots of unity form a cyclic subgroup

All I recall about $\pi$, is that it is the torus and that it is a subgroup of $\mathbb C^\times$. I read about the proof that forms a cyclic subgroup of $\mathbb C^\times$, but not sure about this ...
3
votes
2answers
61 views

Prove that the function $f:G \rightarrow G$ given by $f(a^i)=b^i$ is an automorphism of $G$

Assume that $a$ and $b$ are both generators of the cyclic group $G$, so that $G=<a>$ and $G=<b>$. Prove that the function $f:G \rightarrow G$ given by $f(a^i)=b^i$ is an automorphism of ...
7
votes
8answers
211 views

Proof of Divisibility of $n(n^2+20)$ by 48.

This is a question from Bangladesh National Math Olympiad 2013 - Junior Category that still haunts me a lot. I want to find an answer to this question. Please prove this. If $n$ is an even ...
0
votes
1answer
35 views

Proving addition and multiplication

(1)Show that addition and multiplication mod n are associative operations. (2)Show that there are both an additive and a multiplicative identity. (3)Show that multiplication distributes over ...
2
votes
3answers
78 views

$n$-abelian Groups

Show that $(xy)^n=x^ny^n$ if $xy=yx$. I assume I will need 3 different cases: $n < 0$, $n=0$, and $n > 0$. For the $n > 0$ case, can I use induction? For the base case I'll show that ...
2
votes
3answers
101 views

Prove that $x^{−1}Hx $ is a subgroup of G [closed]

Let $H$ be a subgroup of a group $G$ and, for $x\in G$, let $x^{-1}ax$ denote the set $\{x^{−1}ax : a\in H\}$. Prove that $x^{−1}Hx$ is a subgroup of $G$.
4
votes
0answers
66 views

Suppose $G=\{a_1,\dots a_n\}$ is a finite abelian group and $a^2\neq e$ for all nonidentity elements. What is the product $a_1a_2\dots a_n$?

Let $G$ be a finite abelian group such that for all $a \in G$, $a\ne e$, we have $a^2\ne e$. If $a_1, a_2, \ldots, a_n$ are all the elements of $G$ with no repetitions, what is the product $a_1 a_2 ...
9
votes
1answer
185 views

Prove that the multiplicative groups $\mathbb{R} - \{0\}$ and $\mathbb{C} - \{0\}$ are not isomorphic.

Is my proof correct? I have made use of the fact isomorphism preserves order of elements, which I proved couple of exercises back. I am also interested in other ways of proving it. Is there a more ...
2
votes
0answers
124 views

Show that an element has order $p$ in $S_n$ iff its cycle decomposition is a product of commuting $p$-cycles.

I would like to know if my proof below is correct. Problem Let $p$ be a prime. Show that an element has order $p$ in $S_n$ iff its cycle decomposition is a product of commuting $p$-cycles. Show by ...
1
vote
1answer
50 views

If $\tau = (1,2)(3,4,5)$ determine whether there is a $n$-cycle $\sigma$ ($n\geq 5$) with $\tau = \sigma^k$ for some integer $k$.

I would like to know if my proof below is correct. Problem If $\tau = (1,2)(3,4,5)$ determine whether there is a $n$-cycle $\sigma$ ($n\geq 5$) with $\tau = \sigma^k$ for some integer $k$. ...
3
votes
3answers
92 views

If $\Omega = \{1,2,3,\ldots,\}$, then $S_{\Omega}$ is an infinite group.

I would like to know if my proof below is correct. I do not have issues proving that $S_{\Omega}$ is a group; what I am not sure is whether my proof that $\vert S_{\Omega} \vert = \infty$ is correct. ...
12
votes
4answers
269 views

Tips for writing proofs

When writing formal proofs in abstract and linear algebra, what kind of jargon is useful for conveying solutions effectively to others? In general, how should one go about structuring a formal proof ...
0
votes
1answer
31 views

Fields and irreducible polynomial of $p^n$ degree

Let $K$ be a field of $p$ elements. Let $f(x) \in K [x]$ be an irreducible polynomial of degree $n$. Prove that the field $K[x]/(f(x))$ has $p^n$ elements. By given theorem, let $K$ be a field, ...
1
vote
1answer
70 views

What ring is the quotient $\mathbb{Z}[\sqrt{-11}]/(3,1+\sqrt{-11})$ isomorphic to?

Could anyone help me with this question? I've the feeling that the answer is $\mathbb{Z}/3\mathbb{Z}$, but I'm not sure at all and above all I don't know how to prove it. Thanks
3
votes
2answers
350 views

Proof GL2(Z/2Z) is isomorphic to S3

I'm asked to show that $G=GL_2(\Bbb Z/2\Bbb Z)$ is isomorphic to $S_3$. I have few ideas but I don't manage to put them all together in order to obtain a satisying answer. I first tried using Cayley's ...
2
votes
1answer
50 views

Isomorphism & Quotient Ring

Consider the function $\varphi$ from $R=\{a+bi|a,b\in\mathbb{Z}\}$ to $\mathbb{Z}/p\mathbb{Z}$ with $p=4m+1$,defined by $\varphi(a+bi)=[a+bt]p$ where $[j]p$ denotes the residue class of the integer $j ...
2
votes
1answer
194 views

Proof that there is only one homomorphism from Z to Z/nZ

Could anyone help me (even just a start) to prove this ? Homomorphism is a new notion for me and I have to confess that I am a bit lost, I don't know how to start. Thanks in advance
0
votes
1answer
84 views

Show a subring contains certain elements.

Show that the set of all real numbers of the form $a_0 + a_1\pi + a_2\pi^2 +\cdots+ a_n\pi^n$ with $n≥0$ and $a_i ∈ \mathbb{Z}$ is a subring of $R$ that contains $\mathbb{Z}$ and $\pi$. ...
1
vote
2answers
89 views

Is it ok to prove a subset of a group is an abelian group this way?

I'll admit from the start I'm being lazy, but all the same it makes thing's neater in my opinion - if it's valid. Now it's known that if we have a group $G$ such that $g^2=e,\ \ \forall g \in G$ then ...
0
votes
1answer
33 views

Question about definition of generator for dihedral groups?

Dummit and Foote mention that a relation for the dihedral group is $rs = sr^{-1}$. Now, I have interpreted the statement to mean $r$ is a rotation of $\frac{2\pi}{n}$ radians and $s$ is an ...
3
votes
1answer
58 views

Rephrase the proof “For all odd n, there exists a Group G …” [duplicate]

I am trying to construct a proof and would like to know if I have started it correctly. The proof is as follows. "Prove that for every odd integer n, there is a group with exactly n elements of order ...
1
vote
1answer
76 views

Proof Checking and input: Generators of $\mathbb{Z}_{pq}$

I'm self-studying abstract algebra (slowly but surely), and I have a question about my answer to the following prompt: Problem statement: Show that there are $(q-1)(p-1)$ generators of the group ...
1
vote
1answer
97 views

If $\Omega = \{1,2,3,\dots\}$ then $S_\Omega$ is an infinite group.

I don't think I quite get what the question is looking for. I wonder if anyone could point my attempt to the right track? Prove that if $\Omega = \{1,2,3,\dots\}$ then $S_\Omega$ is an infinite ...
1
vote
1answer
52 views

If $p$ is irreducible and $p \not \mid a$, then $\text{gcd}(p,a)=\pm 1$.

I will be taking a Rings and Fields course in the Fall, so I figured I would read ahead in the textbook (A First Course in Abstract Algebra, by Anderson and Feil) to prepare. Recall the following ...
1
vote
2answers
556 views

Proof Help: In a group $G$, there exists a $g$ such that $g^2 = e$ [duplicate]

I'm working through an Abstract Algebra book to teach myself, and came across the problem: Prove: If $G$ is a finite group of even order, then there exists a $g\in G$ such that $g^2 = e$ and $g ...
-4
votes
2answers
92 views

Help with Theorem III.3.11 in Hungerford's algebra book

I need help to prove part (i) of this theorem which I couldn't prove. Any help would be appreciated. Thanks in advance.
6
votes
4answers
165 views

Any commutative associative operation can be extended to a function on nonempty finite sets

This is a fact we use very frequently in general mathematics when we write such notations as $1+2+3+4$: since we know that $+$ is commutative and associative, we can just "drop the parentheses" and ...
0
votes
2answers
63 views

Help in a proof of a result in Hungerford's book

I need help to proof the last part of this corollary: I didn't understand the part (IV) because the author proves just the canonical projection case and the statement says "every nonzero ...
3
votes
1answer
95 views

$AX=C$: An Inconsistent Linear Equation [duplicate]

Question: Let $A \in M_{n\times n}(F)$. Suppose that the system of linear equations $AX = B$ has more than one solution. Prove that there is a column $C \in F^n$ such that the system of linear ...
2
votes
1answer
85 views

how to show associativity of multiplication for not just 3 operands but for n operands

ie Id like to show a(bc)=(ab)c but for any n operands eg abcdefg=gfdcabe etc I can see this is very intuitive that this should be true for all n operands, but as a logical exercise I would like to ...
3
votes
1answer
189 views

Finite Abelian groups, G, H, K: $G \times H \cong G\times K$ then $H\cong K$

Let $G,H,$ and $K$ be finite abelian groups then if $G \times H \cong G\times K$ then $H\cong K$. I am trying to use the fundamental theorem for abelian groups to solve this, it is clear intuitively ...
3
votes
2answers
320 views

Prove the third isomorphism theorem

I'm trying to prove the third Isomorphism theorem as stated below Theorem. Let $G$ be a group, $K$ and $N$ are normal subgroups of $G$ with $K⊆N$. Then $$(G⁄K)⁄(N⁄K)≅G⁄N.$$ I look up for some ...
4
votes
2answers
1k views

Prove that such an inverse is unique

Given $z$ is a non zero complex number, we define a new complex number $z^{-1}$ , called $z$ inverse to have the property that $z\cdot z^{-1} = 1$ $z^{-1}$ is also often written as $1/z$
1
vote
1answer
206 views

What is the difference between a binary relation and an equivalence class?

Is an equivalence class essentially a binary relation whose elements have an equivalence relation?
0
votes
2answers
705 views

Prove that any group $G$ of order $p^2$ is abelian, where $p$ is a prime number [duplicate]

Possible Duplicate: Showing non-cyclic group with $p^2$ elements is Abelian "Let $p$ be a prime number. Prove that any group $G$ of order $p^2$ is abelian. You may assume the fact that the ...
1
vote
1answer
50 views

$H_1 ,H_2 \unlhd \, G$ with $H_1 \cap H_2 = \{1_G\} $. Prove every two elements in $H_1, H_2$ commute

This is the proof, which I mostly understand except for one bit: You have $h_1 \in H_1$ and $h_2 \in H_2$. We also have $h_1^{-1}(h_2^{-1}h_1h_2) \in H_1$, because $h_2^{-1}h_1h_2 \in ...
0
votes
3answers
284 views

Describe geometrically the elements of the equivalence classes

After showing that the relation $R$ defined on $\mathbb R\times \mathbb R$ by $((a,b), (c,d))\in R$ if $|a|+|b|=|c|+|d|$ is an equivalence relation (I already did), how can I describe geometrically ...
0
votes
2answers
147 views

Proof of Normal subgroup in symmetry group

Prove that $A_n$ is a normal subgroup of $S_n$ Proof A subgroup is normal if its left cosets are the same as its right cosets. That is, we want to show that for s in ...
2
votes
3answers
312 views

Resources for Proof practice

I am a Math fan and a self-learner. I try to look at least once a day at a Linear Algebra or Calculus problem to keep myself in shape and to learn. I also like Analysis, Abstract Algebra and Discrete ...
3
votes
2answers
2k views

Proof of $\gcd(a,b)=ax+by$

Here is my proof of $\gcd(a,b)=ax+by$ for $a, b, x, y \in \mathbb{Z}$. Am I doing something wrong? Are there easier proofs? $a,b \in \mathbb{Z}, g=\gcd(a,b)$ and suppose $g \neq ax + by$. Let $c$ be ...
2
votes
3answers
361 views

How to prove that $z\cdot\text{gcd}(a,b)=\text{gcd}(za,zb)$

I need to prove that $z \cdot \text{gcd}(a,b)=\text{gcd}(za,zb)$. I tried a lot, for example, looking at set of common divisors of the two sides, but I can't conclude anything from that. Can you ...
1
vote
2answers
95 views

Is using the - symbol with the Associative Law of multiplication invalid?

I was trying to prove that $-(x + y) = -x - y$ and as you can see in the image below, I took the liberty of using the $-$ symbol as a number and applying the associative law with it. Is it kosher in ...