= Conjugates of a proper subgroup do not cover a finite group
{title2=$|\bigcup_g gBg^{-1}|\leq |G|-[G:B]+1<|G|$}
For a proper <subgroup> $B$ of a <finite group> $G$, the number $m$ of its conjugates is $[G:N_G(B)]\leq[G:B]$, because its <normalizer> contains $B$. All conjugates contain the identity, so their union has size at most $1+m(|B|-1)\leq|G|-[G:B]+1<|G|$. Thus some element is not conjugate to any element of $B$. Finiteness is essential to this cardinal counting argument.
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