Tessellate S.T.E.M.S - Mathematics - College - Set 3 - Subjective Problem

Let $G$ be a group, $H$ be a subgroup of $G$ such that $G/H$ is finite.

Do there exist $a_1,a_2, \dots ,a_n \in G$ such that

$a_1H, a_2H, \dots , a_nH$ form all the left cosets of $H$ , and
$Ha_1, Ha_2, \dots , Ha_n$ form all the right cosets of $H$ ?

#Algebra

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Comments

Yes

anuj malik - 2 years, 5 months ago

Consider G to be {a1,a2......an} and H = {e} , the trivial subgroup containing only the identity element of the group . Then index of H in G is n . So each of the left cosets are unique. This holds true for Right Cosets as well because it is given that G/H exists which implies H must be normal in G

Arghyadeep Chatterjee - 1 year, 9 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$