3
votes
3answers
47 views

Group of even order must contain $a:a=a^{-1}$ $ (a\not = e)$ [duplicate]

Let $G$ be a finite group. If the order of $G$ is even, prove that there is at least one element $a$ in $G$ such that $a\not= e$ and $a=a^{-1}$. Here's my idea: Suppose $\{x_1,\cdots,x_n\}$ is ...
0
votes
2answers
41 views

Prove that the order of $U(n)$ is even when $n>2$.

I'm trying to provide a solution to the following claim: "The order of $U(n)$ is even when $n>2$." Note: here, $U(n)$ is the set of all positive elements that are less than and relatively prime ...
0
votes
1answer
29 views

Converse of DIC

I am looking to prove the converse of the Divisibility of Integer Combination. I know how to prove the contrapositive of this statement but not the converse ... any help? Below is my attempt at ...
0
votes
1answer
32 views

If $|G|=p^n$ for prime $p$, then $|\mathcal{Z} (G)|\neq p^{n-1}$

I am trying to prove the following: Let $|G|=p^n$ for $n\geq 1$ and $p$ prime. Prove that $|\mathcal{Z} (G)|\neq p^{n-1}$. Here is what I have so far: Suppose, to the contrary, that $|\mathcal{Z} ...
3
votes
2answers
39 views

Disjoint Cycles and Supports

I am working though an Introduction to the theory of Groups. I have come the following exercise: "Let $\alpha = \begin{pmatrix} i_1 & i_2 & \cdots & i_r \end{pmatrix}$ and $\beta = ...
0
votes
1answer
36 views

Equivalence relations and power sets.

Let $\mathcal{A}$ be the class of all sets and define the relation $R$ on $\mathcal{A}$ as: $A\space R\space B$ iff there is a bijective function $f:A \to B$. Prove that $R$ is an equivalence relation ...
2
votes
3answers
176 views

Proving units in a ring

Suppose $R$ is a ring with no zero divisors and with identity $1_R$ not equal to $0_R$. Suppose that $a,b$ are in $R$ and that $ab$ is a unit. Prove that $b$ is a unit. My thoughts: I know a ...
1
vote
1answer
55 views

Confusion with application of butterfly lemma in Lang's Algebra

In Lemma 3.3 of Serge Lang's Algebra, the so-called Butterfly Lemma is proved: And then Lang proceeds to prove Schreier's Refinement Theorem (highlight mine): In the proof of Schreier's theorem, ...
0
votes
1answer
31 views

Eisenstein's criterion pf

I know that 'Eisenstein's criterion'. I know that pf of state "(NOT $p$|$a_{n}$), [$P|a_i$ for ($0\le i\le n-1$)], (NOT $p^2$|$a_0$)". I know regular way. but I hope to Second pf way. $\;$ $\;$ ...
0
votes
1answer
46 views

$f(x)$ is irreducible and primitive over $\Bbb Q$, then $f(x)$ is irreducible over $\Bbb Z$

show that $f(x) (\in \Bbb Z[x])$ is irreducible and primitive over $\Bbb Q$, then $f(x)$ is irreducible over $\Bbb Z$. How pf it? I tried it. MY pf) Suppose that $f(x)$ is reducible over $\Bbb Z$. ...
0
votes
1answer
31 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
39 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
31 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
18 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
41 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
40 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
75 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
34 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
64 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
254 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
48 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
88 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 ...
1
vote
3answers
114 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
74 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 ...
12
votes
2answers
306 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
144 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
66 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 $k\in\mathbb N$.

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
110 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. ...
13
votes
4answers
343 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
32 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
72 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
479 views

Prove $GL_2(\mathbb{Z}/2\mathbb{Z})$ is isomorphic to $S_3$

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
55 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
253 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
105 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
39 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
60 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
83 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
99 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
53 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
689 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
99 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
185 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
66 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
97 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
86 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
203 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
385 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 ...