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Abstract algebra Answers
Abstract algebra
asked 2021-02-27
In the froup
\(\displaystyle{Z}_{{12}}\)
, find |a|, |b|, and |a+b|
a=5, b=4
Abstract algebra
asked 2021-02-27
Prove the following.
(1) Z ∗ 5 is a cyclic group.
(2) Z ∗ 8 is not a cyclic group.
Abstract algebra
asked 2021-02-26
Let H be a normal subgroup of a group G, and let m = (G : H). Show that
\(a^(m)inH\)
for every
\(a in G\)
Abstract algebra
asked 2021-02-25
If U is a set, let
\(\displaystyle{G}={\left\lbrace{X}{\mid}{X}\subseteq{U}\right\rbrace}\)
. Show that G is an abelian group under the operation \oplus defined by
\(\displaystyle{X}\oplus{Y}={\left({\frac{{{x}}}{{{y}}}}\right)}\cup{\left({\frac{{{y}}}{{{x}}}}\right)}\)
Abstract algebra
asked 2021-02-25
Let (Z,+) be a group of integers and (E,+) be a group of even integers. Find and prove if there exist an isomorphism between them.
Abstract algebra
asked 2021-02-15
Let F be a field, and
\(\displaystyle{p}{\left({x}\right)}\in{F}{\left[{x}\right]}\)
an irreducible polynomial of degreed. Prove that every coset of
\(\displaystyle{F}\frac{{{x}}}{{{p}}}\)
can be represented by unique polynomial of degree stroctly less than d. and moreover tha these are all distinct. Prove that if F has q elements,
\(\displaystyle{F}\frac{{{x}}}{{{p}}}\)
has
\(\displaystyle{q}^{{d}}\)
elements.
Abstract algebra
asked 2021-02-11
Let a,b be coprime inegers. Prove that every integer x>ab-a-b can be written as na-mb where n,m where are non-negative inegers. Prove that ab-a-b connot be expressed ib this form.
Abstract algebra
asked 2021-02-09
Show that the prime subfield of a field of characteristic p is ringisomorphic to Zp and that the prime subfield of a field of characteristic 0 is ring-isomorphic to Q.
Abstract algebra
asked 2021-02-08
Prove that if "a" is the only elemnt of order 2 in a group, then "a" lies in the center of the group.
Abstract algebra
asked 2021-02-06
What is Triangularization of linear operator
Abstract algebra
asked 2021-02-05
Let RR sube K be a field extension of degree 2, and prove that K ~= CC. Prove that there is no field extension RR sube K of degree 3.
Abstract algebra
asked 2021-02-02
Let
\(\displaystyle{G}={S}_{{3}}{\quad\text{and}\quad}{H}={\left\lbrace{\left({1}\right)}{\left({2}\right)}{\left({3}\right)},{\left({12}\right)}{\left({3}\right)}\right\rbrace}\)
. Find the left cosets of
\(\displaystyle{H}\in{G}\)
.
Abstract algebra
asked 2021-01-31
Let
\(\displaystyle{H}={\left\lbrace\sigma\in{S}_{{5}}{\mid}\sigma{94}\right)}={4}\rbrace\)
Show that
\(\displaystyle{H}\le{S}_{{5}}\)
Abstract algebra
asked 2021-01-31
In an abstract algebra equation about groups, is "taking the inverse of both sides of an equation" an acceptable operation? I know you can right/left multiply equations by elements of the group, but was wondering if one can just take the inverse of both sides?
Abstract algebra
asked 2021-01-27
Let G be a group of order
\(\displaystyle{p}^{{m}}\)
where p is prime number and m is a positive integer. Show that G contains an element of order p.
Abstract algebra
asked 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
\(\displaystyle\frac{{G}}{{H}}\)
is cyclic.
Definition: A subgroup H of a group is said to be a normal subgroup of G it for all
\(\displaystyle{a}\in{G}\)
, aH = Ha
Definition: Suppose G is group, and H a normal subgruop og G. THe froup consisting of the set
\(\displaystyle\frac{{G}}{{H}}\)
with operation defined by (aH)(bH)-(ab)H is called the quotient of G by H.
Abstract algebra
asked 2021-01-23
what is abelin group?
Abstract algebra
asked 2021-01-19
In group theory (abstract algebra), is there a special name given either to the group, or the elements themselves, if
\(\displaystyle{x}^{{2}}={e}\)
for all x?
Abstract algebra
asked 2021-01-15
In the froup
\(\displaystyle{Z}_{{12}}\)
, find |a|, |b|, and |a+b|
a=6, b=2
Abstract algebra
asked 2021-01-08
Find the inverse of
\(\displaystyle{x}+{1}\in\mathbb{Q}\frac{{{x}}}{{{x}^{{3}}-{2}}}\)
. Explain why this is the same as finding the inverse of
\(\displaystyle{\sqrt[{{3}}]{{{2}}}}\in\mathbb{R}\)
.
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