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

Question
Abstract algebra
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 \(\displaystyle{b}\in{a}{H}\).

Answers (1)

2020-12-16
Suppose
\(\displaystyle{a}{H}\cap{b}{H}\ne\emptyset\)
Let \(\displaystyle{x}\in{a}{H}\cap{b}{H}\), then there exist \(\displaystyle{h}_{{1}},{h}_{{2}}\in{H}\)
Such that,
\(\displaystyle{x}={a}{h}_{{1}}&{x}={b}{h}_{{2}}\)
\(\displaystyle{a}={x}{{h}_{{1}}^{{-{{1}}}}}&{b}={x}{{h}_{{2}}^{{-{{1}}}}}\)
\(\displaystyle{a}={b}{h}_{{2}}{{h}_{{1}}^{{-{{1}}}}}\)
\(\displaystyle{a}{H}={b}{h}_{{2}}{{h}_{{1}}^{{-{{1}}}}}{H}={b}{H}\)
(since \(\displaystyle{a}{H}={H}{\quad\text{if}\quad}{a}\in{H}\))
Now, if \(\displaystyle{a}\in{b}{H}\), using above theorem
Since \(\displaystyle{b}\in{b}{H}\), then \(\displaystyle{a}{H}={b}{H}\Rightarrow{b}\in{a}{H}\)
\(\displaystyle{a}\in{H}{\quad\text{and}\quad}{a}\in{a}{H}\Rightarrow{a}\in{a}{H}\cap{b}{H}\)
\(\displaystyle\Rightarrow{a}{H}\cap{b}{H}\ne{0}\)
\(\displaystyle\Rightarrow{S}{o}{a}{H}\cap{b}{H}={0}\)
\(\displaystyle\Rightarrow{b}\in{a}{H}\)
Hence G be a group, H a subgroup of G, and a and b elements of G. If \(\displaystyle{a}\in{b}{H}\) then \(\displaystyle{b}\in{a}{H}\).
0

Relevant Questions

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 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 \(a in G\)
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 2021-01-31
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?
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 2021-02-15
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.
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 \(\displaystyle{X}\oplus{Y}={\left({\frac{{{x}}}{{{y}}}}\right)}\cup{\left({\frac{{{y}}}{{{x}}}}\right)}\)
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.
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 2020-12-27
Use Principle of MI to verify
(i) If \(\displaystyle{n}\in\mathbb{Z}\) is a positive ineger then \(\displaystyle{2}^{{n}}{3}^{{{x}{n}}}-{1}\) divisible by 17.
(ii) For all positive integers \(\displaystyle{n}\ge{5}\),
\(\displaystyle{2}^{{k}}{>}{k}^{{2}}\)
...