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Why is $G/H$ smaller than $G$ when $G$ is finite and $H \neq \{e\}$? Reading Chapter 9 of Gallian on Normal Subgroups and Factor Groups when this assertion is made referring to why factor groups are important.

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  • $\begingroup$ The cosets of $H$ are disjoint, and each has the same number of elements as $H$. The elements of $G/H$ are just these cosets. $\endgroup$ – Thomas Andrews May 14 '14 at 0:31
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One may prove that $$ \left|G/H\right|=|G|/|H| $$ so $|H|>1$ implies $|G/H|<|G|$.

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An argument similar to Brian's ( it may be helpful to get different perspectives.) is that the cosets form a partition of the group. By a counting argument, there can be at most |G| cosets in $G/H$ (maybe assume normality of H to avoid well-definedness issues in G/H ).

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