# Tagged Questions

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### If $N$ normal subgroup of $G$ and $M$ normal subgroup of $G$ prove $MN$ is a subgroup [closed]

If $N\vartriangleleft G$ and $M\vartriangleleft G$ and $MN = \{mn | m \in M, n \in N\}$, prove that $MN$ is a subgroup of $G$ and that $MN\vartriangleleft G$
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### Show that lcm$(a,b)= ab$ if and only if gcd$(a,b)=1$

Not sure how to begin. If gcd$(a,b)=1$ what can I deduce from that?
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### My first proof related to subspaces (vector spaces). Please comment.

What do you think about my first proof which deals with subspaces? Theorem An intersection of subspaces is a subspace. Preliminaries Corresponding to the notation in Wikipedia, symbols for vectors ...
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### Fields - Proof that every multiple of zero equals zero

This is one of my first proofs about fields. Please feed back and criticise in every way (including style and details). Let $(F, +, \cdot)$ be a field. Non-trivially, $\textit{associativity}$ implies ...
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### Proof that group is commutative if every element is its inverse (feedback wanted)

This is one of my first proofs about groups. Please feed back and criticise in every way (including style & language). Axiom names (see Wikipedia) are italicised. $e$ denotes the identity element. ...
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### Groups - Prove that every element equals inverse of inverse of element

This is my first proof about groups. Please feed back and criticise in every way (including style & language). Axiom names (see Wikipedia) are italicised. We use $^{-1}$ to denote inverse ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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. $\;$ $\;$ ...
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### $f(x)$ 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 $\Bbb Z$. ...
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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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### 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 ...
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### 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 ...
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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 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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### 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. ...
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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 ...
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### 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, ...
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### 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
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### 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 ...