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

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 $$\displaystyle{b}\in{a}{H}$$.

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