# Tagged Questions

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### 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 ...
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### 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?
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### 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 $c(x) = (x^2-1)^4(x^2+3x+2)$. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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) $\ x+y=y+x$ ...
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### 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).
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $k$. ...
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### 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 ...
(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 ...
### $\mathbb{C}[x,y]/(x^2+y^2+1)$ 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, $c \in \mathbb{F}$. ...