In the froup Z_12, find |a|, |b|, and |a+b| a=6, b=2

Question
Abstract algebra
asked 2021-01-15
In the froup \(\displaystyle{Z}_{{12}}\), find |a|, |b|, and |a+b|
a=6, b=2

Answers (1)

2021-01-16
Consider the provided qustion,
If a \(\displaystyle\in{Z}_{{n}}\) then \(\displaystyle{\left|{a}\right|}=\frac{{n}}{{{\gcd{{\left({a},{n}\right)}}}}}\)
here given \(\displaystyle{Z}_{{12}}\). So, n=12
Given a = 6, b = 2
So, a + b = 6 + 2 = 8
\(\displaystyle{\left|{a}\right|}={\left|{6}\right|}=\frac{{12}}{{{\gcd{{\left({6},{12}\right)}}}}}=\frac{{12}}{{6}}={2}\)
\(\displaystyle{\left|{b}\right|}={\left|{2}\right|}=\frac{{12}}{{{\gcd{{\left({2},{12}\right)}}}}}=\frac{{12}}{{2}}={6}\)
\(\displaystyle{\left|{a}+\right|}={\left|{8}\right|}=\frac{{12}}{{{\gcd{{\left({8},{12}\right)}}}}}=\frac{{12}}{{4}}={3}\)
0

Relevant Questions

asked 2021-02-27
In the froup \(\displaystyle{Z}_{{12}}\), find |a|, |b|, and |a+b|
a=5, b=4
asked 2020-12-30
In the froup \(\displaystyle{Z}_{{12}}\), find |a|, |b|, and |a+b|
a=3, b=8
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.
asked 2020-11-11
An nth root of unity epsilon is an element such that \(\displaystyle\epsilon^{{n}}={1}\). We say that epsilon is primitive if every nth root of unity is \(\displaystyle\epsilon^{{k}}\) for some k. Show that there are primitive nth roots of unity \(\displaystyle\epsilon_{{n}}\in\mathbb{C}\) for all n, and find the degree of \(\displaystyle\mathbb{Q}\rightarrow\mathbb{Q}{\left(\epsilon_{{n}}\right)}\) for \(\displaystyle{1}\le{n}\le{6}\)
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 \(\displaystyle{H}\in{G}\).
asked 2020-12-15
Suppose G is a group, H a subgroup of G, and a and b elements of G. If \(\displaystyle{a}\in{b}{H}\) then \(\displaystyle{b}\in{a}{H}\).
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 \(\displaystyle{\sqrt[{{3}}]{{{2}}}}\in\mathbb{R}\).
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.
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?
asked 2021-01-05
Let a,b be coprime integers. Prove that every integer x>ab-a-b can be written as na+mb where n,m are non negative integers. Prove that ab-a-b cannot be expressed in this form.
...