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Assume that G is a group of order $p^{m}$ , where m is a positive integer and p is a prime number. Display the presence of an element of order p in G.

nagasenaz 2021-01-25

Suppose G is a group and H is a normal subgroup of G. Prove or disprove ass appropirate. If G is cyclic, then $\frac{G}{H}$ is cyclic.
Definition: A subgroup H of a group is said to be a normal subgroup of G it for all $a \in G$ , aH = Ha
Definition: Suppose G is group, and H a normal subgruop og G. THe froup consisting of the set $\frac{G}{H}$ with operation defined by (aH)(bH)-(ab)H is called the quotient of G by H.

Cem Hayes 2021-01-23

what is abelin group?

arenceabigns 2021-01-19

In group theory (abstract algebra), is there a special name given either to the group, or the elements themselves, if $x^{2} = e$ for all x?

necessaryh 2021-01-15

In the froup $Z_{12}$ , find |a|, |b|, and |a+b|
a=6, b=2

foass77W 2021-01-08

Find the inverse of $x + 1 \in Q \frac{x}{x^{3} - 2}$ . Explain why this is the same as finding the inverse of $\sqrt[3]{2} \in R$ .

Lewis Harvey 2021-01-06

Prove that a group of even order must have an element of order 2.

Joni Kenny 2021-01-05

Let a,b be coprime integers. Prove that every integer $x > a b - a - b$ can be written as $n a + m b$ where n,m are non negative integers. Prove that $a b - a - b$ cannot be expressed in this form.

Dolly Robinson 2021-01-04

Let F be a field and consider the ring of polynominals in two variables over F,F[x,y]. Prove that the functions sending a polyomial f(x,y) to its degree in x, its degree in y, and its total degree (i.e, the highest $i + j$ where $x^{i} y^{i}$ appears with a nonzero coefficient) all fail o be norm making F[x,y] a Euclidean domain.

texelaare 2020-12-30

In the froup $Z_{12}$ , find $| a |, | b |$ , and $| a + b |$
$a = 3, b = 8$

hexacordoK 2020-12-27

Use Principle of MI to verify
(i) If $n \in Z$ is a positive ineger then $2^{n} 3^{x n} - 1$ divisible by 17.
(ii) For all positive integers $n \geq 5$ ,
$2^{k} > k^{2}$

Rui Baldwin 2020-12-15

Let F be i field with subfields K,L. Prove that there is a largest subfield of F contained in both K and L, and a smallest subfield of containung both K and L.

nicekikah 2020-12-15

How many subgroups of order 4 does $D_{4}$ have?

Harlen Pritchard 2020-12-15

Suppose G is a group, H a subgroup of G, and a and b elements of G. If $a \in b H$ then $b \in a H$ .

iohanetc 2020-11-29

How to find the value of U(n) on abstract algebra
Example U(8) = {1,3,5,7}

Khadija Wells 2020-11-20

Show that an element and its inverse have the same order in any group.

mattgondek4 2020-11-11

An nth root of unity epsilon is an element such that $ϵ^{n} = 1$ . We say that epsilon is primitive if every nth root of unity is $ϵ^{k}$ for some k. Show that there are primitive nth roots of unity $ϵ_{n} \in C$ for all n, and find the degree of $Q \to Q (ϵ_{n})$ for $1 \leq n \leq 6$

rocedwrp 2020-11-10

Find th eminimal polynomial of $\sqrt{2} + \sqrt{3}$ over $Q$

Mylo O'Moore 2020-10-23

Write down a definition of a subfield. Prove that the intersection of a set of subfields of a field F is again a field

Isa Trevino 2020-10-23

F should be a field. Show that there are infinitely many monic polynomials that are irreducible.

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