Expert answers to "Suppose G is a group, H a subgroup..."

College
Algebra
Abstract algebra

Harlen Pritchard

Answered question

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 $b \in a H$ .

Answer & Explanation

opsadnojD

Skilled2020-12-16Added 95 answers

Suppose
$a H \cap b H \neq \emptyset$
Let $x \in a H \cap b H$ , then there exist $h_{1}, h_{2} \in H$
Such that,
$x = a h_{1} and x = b h_{2}$
$a = x h_{1}^{- 1} and b = x h_{2}^{- 1}$
$a = b h_{2} h_{1}^{- 1}$
$a H = b h_{2} h_{1}^{-} 1 H = b H$
(since $a H = H if a \in H$ )
Now, if $a \in b H$ , using above theorem
Since $b \in b H$ , then $a H = b H \Rightarrow b \in a H$
$a \in H and a \in a H \Rightarrow a \in a H \cap b H$
$\Rightarrow a H \cap b H \neq 0$
$\Rightarrow S o a H \cap b H = 0$
$\Rightarrow b \in a H$
Hence G be a group, H a subgroup of G, and a and b elements of G. If $a \in b H$ then $b \in a H$ .

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