0
votes
1answer
15 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 ...
0
votes
0answers
15 views

Order of a Group Help Abstract [duplicate]

Is the order of the Heisenberg group infinite since H = 1 a b 0 1 c 0 0 1 under matrix multiplication where a,b,c are real numbers? How would I formally state this?
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 $c(x) = (x^2-1)^4(x^2+3x+2)$. ...
0
votes
1answer
31 views

Normal Subgroups and Isomorphisms Help

Prove or give a counterexample: If $H, K$ are normal subgroups to $G$ and $G/H$ is isomorphic to $G/K$, then $H$ is isomorphic to $K$. proof: Let $G$ be the Klein-4 Group ($V$), $H = \langle ...
1
vote
2answers
22 views

Normal Subgroups Proof Help Abstract

Prove or disprove the following assertion. The set of all nonzero scalars matrices is a normal subgroup of $GL_2(\mathbb{R})$. Proof: Let $I$ be the identity matrix. Consider the scalar matrix $sI$ ...
1
vote
2answers
22 views

Proof of Conjugate Subgroup Isomorphism

Let $G$ be a group, and let $H$ be a subgroup of $G$. Prove that if $a$ is an element of $G$, then the subset $aHa^{-1} = \{g ∈ G | g = aha^-1 \text{ for some } h \in H\}$ is a subgroup of $G$ that is ...
0
votes
0answers
10 views

Preservation of a map

There is a map from Z(mod12) to Z(mod4) defined by f(x)=3x. The thought I had was this. Say you have [a],[b] that are in Z(mod12). Would f([a][b])=f([a])f([b])? So you basically view this as a ...
0
votes
3answers
42 views

Abstract Algebra Quotient Group and Isomorphism Proof Help

If $G$ is an abelian group, $S = \{ y \in G \; : \; y = x^2\; \exists x \in G\}$, and $T = \{ a \in G \; :\; a^2 = e\}$, then $G/T$ is isomorphic to $S$. Proof: Let $G$ be an abelian group, $S = \{ ...
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 ...
0
votes
2answers
52 views

How to show that a regular pentagon can't have all coordinates rational

This is pretty straightforward if we're allowed to use trigonometry, so I guess my question is Are there any nice (trigonometry-less) proofs of the fact that a regular pentagon in the plane must ...
0
votes
0answers
49 views

Modular arithmatic

Suppose that f : Zmod12 -> Zmod4 is defined by f [x] mod 12 = [3x]mod 4 where the subscript indicates the appropriate modular arithmetic. (A) IS f surjective? (B) is f injective? (C) let [a], [b] be ...
0
votes
2answers
29 views

Abstract Algebra Subgroup Proof Help

Show that if N is a normal subgroup of G and |N| = 2, then N is a subgroup of Z(G). proof: Let N be a normal subgroup of G. Then N is a subgroup of G and g is in G. So gN = Ng for all g in G. Suppose ...
0
votes
1answer
32 views

Isomorphism of direct product of semigroups

I would appreciate some help with the following problem. Consider four semigroups $A,B,C,D$. I was able to prove that $A\cong C\wedge B\cong D$ implies $A\times B\cong C\times D$. But does also ...
0
votes
2answers
39 views

Is this set a subring of $\mathbb{Z}\times\mathbb{Z}$?

Is the set $S = \{(x,-x) : x \text{ is an integer}\}$ a subring of $\mathbb{Z}\times\mathbb{Z}$? I am not sure where to start here. Is $\mathbb{Z}\times\mathbb{Z}$ a matrix? It doesn't seem ...
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
2answers
36 views

Basic proof of statement in abstract algebra?

http://www.proofwiki.org/wiki/Abelian_Quotient_Group The third step (in both proofs) is something I am having trouble seeing. The theorem itself is not difficult to prove, but it is much cleaner this ...
1
vote
1answer
16 views

Every free module is a projective one

I'm trying to understand this proof in Hungerford's book using the universal property of the free modules: In the whole proof I didn't understand just this line, because we can use the uniqueness ...
1
vote
1answer
49 views

subgroup proof.

Prove that if $G$ is an abelian group, then $H =\{ x \in G\mid x^{2} = e \}$ is a subgroup of $G$. I did show that $H$ is close, associative, have identity and inverse element. Then my prof said I ...
0
votes
2answers
54 views

“Classify $\mathbb{Z}_5 \times \mathbb{Z}_4 \times \mathbb{Z}_8 / \langle(1,1,1)\rangle$”

I have a question that says this: Classify $\mathbb{Z}_5 \times \mathbb{Z}_4 \times \mathbb{Z}_8 / \langle(1,1,1)\rangle$ according to the fundamental theorem of finitely generated abelian groups. ...
1
vote
2answers
89 views

Order of Elements in Quotient Groups

Let G be a group, and H a normal subgroup of G. Prove that, for each element a in G, the order of the element Ha in G/H is a divisor of the order of a in G. So I have already done a lot of stuff with ...
0
votes
1answer
132 views

Abstract algebra, prove that $(a^m)^n$ =$ a^{mn}$

Let $a$ be an element of group $G$. For any integers $m,n \in \mathbb{Z}$ ($m,n$ can be positive and negative). Prove that $(a^{m})^{n}=a^{mn}$, then show that $(a^{-1})^{-1} = a$ by using what we ...
2
votes
3answers
48 views

modular arithmetic proof

Suppose $x$, $y$, and $z$ are integers and $x= 3y^2 -z^2$. Prove that $x\not\equiv1\mod4$. My thoughts: So I am not sure the route that can prove this. I am trying to just use the simple stuff to ...
0
votes
1answer
27 views

Greatest common divisor of multiples

What is the GCD of $3 \times 5^2 \times 7^2 \times 11^2$ and $3^2 \times 5^4 \times 11^3$? I can use the euclidean algorithm but is there an easier way to simplify this to make it more simple? If ...
1
vote
3answers
37 views

Proving primes divide each other

Suppose $a,b,p\in\mathbb Z$ with $p$ prime. Prove that if $p\mid a$ and $p \mid a^2 + b^2$, then $p \mid b$. I am starting with the fact that $a=p$t with $t\in\mathbb Z$ and $p= (a^2+b^2)\cdot x$ ...
2
votes
4answers
408 views

Fundamental theorem of arithmetic question

Let $b \in \mathbb{Z} $. Prove that if $p$ is a prime number such that $p | b^2$, then $p|b$. A certain theorem can be used to get this proof set up. I know the general rule that this scenario is ...
2
votes
3answers
79 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
4answers
88 views

Why is $\langle \mathbb{Z}_4, + \rangle$ not isomorphic to $\langle \mathbb{Z}_2 \times \mathbb{Z}_2, + \rangle$?

I'm having some trouble here, specifically with the idea of $\langle \mathbb{Z}_2 \times \mathbb{Z}_2, + \rangle$ as a group. Can anyone help me out with some explanations? Moreover, I generally ...
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 ...
1
vote
3answers
79 views

Define $a\ast b=a+b+5$, and show $(\Bbb Z,\ast)$ is a group.

Let the set $\mathbb Z$ have the operation $*$ defined by $a * b = a + b + 5$ for all $a,b \in\mathbb Z$. Show this is a group. I understand how to prove closure and associativity. For ...
0
votes
2answers
32 views

Groups Math Proof Help

Show that the indicated set $G$ with the specified operation forms a group by showing that the four axioms in the definition of a group are satisfied. $G = \mathbb Z_5$ under addition mod $5$. I ...
2
votes
2answers
34 views

Ordered abelian groups

Consider the following axioms: 1) $\ x+(y+z)=(x+y)+z$ ; $\forall x \forall y \forall z$ 2) $\ x+0=x$ ; $\forall x$ 3) $\forall x$ $ \exists y$ such that $\ x+y=0$ 4) $ \ x+y=y+x$ ...
0
votes
1answer
25 views

Prove If hcf(a,b)|c then then ax+by=c has an integer solution. Where a and b are non-zero integers.

I'm not sure whether to use multiple cases for this particular question (i.e. odd*odd with hcf=1 and odd*even with hcf=1 have integer solutions for x and y).
0
votes
2answers
98 views

How to show that $x$ becomes a root of $p(x)$ in $F[x]/(p(x))$

$F$ is a field, $p(x)$ is irreducible polynomial at $F[X]$. $K=F[X]/\left<p(x)\right>$. For every $a\in F$ we will mark: $\bar{a}=\left<p(x)\right>+a$. Now, the question is: How do I show ...
0
votes
2answers
87 views

Why $I=\left\{p(x)\in \mathbb{Z}\left[X\right]:2\mid p(0)\right\}$ is not a principal ideal? [duplicate]

I saw this question but I still do not understand: What is the difference between ideal and principal ideal? At my homework I had to prove to things about $I=\left\{p(x)\in ...
1
vote
3answers
77 views

For a finite field $F$ of order $n$, all elements are roots of $x^n - x$

I need to prove two things at $F[X]$ but don't know how, Ill glad to get help... $F$ is a finite field. $|F|=n$. We look at $p(x)=x^n-x\in F[X]$ 1. How we can show that every $c\in F$ is a root of ...
0
votes
1answer
37 views

Commutative property based on integral substraction.

I found elementary algebra exercise which I can't resolve. Let algebraic structure $(X,\cdot)$ where $\cdot$ has properties: $$ x\cdot(x\cdot y)=y \\ (y\cdot x)\cdot x=y $$ How to proof that ...
2
votes
2answers
40 views

How does the fundamental theorem of algebra extend to show number (in addition to existence) of roots?

The fundamental theorem of algebra in which we prove a complex polynomial has at least root is clear from the construction of a compact domain and use of the polar coordinate form of complex numbers. ...
3
votes
2answers
70 views

$x$ and $y$ be distinct elements of order $2$ in $G$ that generate $G$. Prove that $G \cong D_{2n}$ [duplicate]

Problem Let $G$ be a finite group and let $x$ and $y$ be distinct elements of order $2$ in $G$ that generate $G$. Prove that $G \cong D_{2n}$, where $\vert xy \vert = n$. Solution We have $x^2 = ...
2
votes
3answers
256 views

Prove that the additive groups $\mathbb{Z}$ and $\mathbb{Q}$ are not isomorphic.

Is my proof below correct? What specific property of rationals did I exploit in my proof? It looks like the property I exploited is the following: Given any positive rational, I can always write it as ...
2
votes
3answers
180 views

Prove that the additive groups $\mathbb{R}$ and $\mathbb{Q}$ are not isomorphic.

Is there a better (or other) way(s) to prove the following statement? Also, the same argument works for multiplicative groups $\mathbb{R}-\{0\}$ and $\mathbb{Q}-\{0\}$, right? Problem Prove that ...
9
votes
1answer
187 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
1answer
66 views

Show that $GL_n(\mathbb{F})$ is a finite group iff $\mathbb{F}$ has a finite number of elements

I would like to know if my proof below is correct. Problem Show that $GL_n(\mathbb{F})$ is a finite group iff $\mathbb{F}$ has a finite number of elements. Solution If $\mathbb{F}$ is a ...
2
votes
0answers
125 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
55 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 $k$. ...
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 ...
4
votes
1answer
77 views

Give the definition of a binary operation for these to be isomorphisms - Fraleigh p. 34 3.18

(1.) In blue: I understand $\phi^{-1}$ is a homomorphism $\iff \phi^{-1}(3a - 1) * \phi^{-1}(3b - 1) = \phi^{-1}((3a - 1) + (3b - 1)).$ But where did the $(3a - 1), (3b - 1)$ crop up from? Why not ...
4
votes
2answers
116 views

$\mathbb{C}[x,y]/(x^2+y^2+1)$ is an integral domain.

I stuck in the following question. Prove that $ \mathbb{C}[x,y]/\langle x^2+y^2+1 \rangle $ is an integral domain, using the following: Let $\mathbb{F}$ be a field, $c \in \mathbb{F} $. ...