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I am trying to get a better understanding for cosets and it seems that since the cosets partition G, there is an equivalence relation on G where any two elements in the same coset of G are equivalent. There are [G:H] cosets of H in G and cosets of H in G look like $gH$ or $g + H$ for $g\in G$. Does the set of cosets of H in G act as a group in a similar fashion to $\mathbb{Z}_n$? Does $[G:H]*(gH) = H$? Does $[G:H]*(g+H) = H$?

mathfan27543
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  • You should choose either the notation $gH$ or $g+H$ for your cosets (usually you use the former for multiplicative groups and the latter for additive groups). The set of cosets do form a group if $H$ is a normal subgroup of $G$. In that case, since the order of $G/H$ is $n$, $(gH)^{[G:H]}=H$. – Sam Weatherhog Oct 24 '16 at 01:34
  • See also http://math.stackexchange.com/questions/573050/if-h-is-a-subgroup-of-g-of-finite-index-n-then-under-what-condition-gn. – lhf Oct 24 '16 at 01:34

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This is true when $H$ is a normal subgroup, but not in general, as this counter-example shows:

Take $G=S_4$, $H=S_3$, $g=(234)$. Then $[G:H]=4$ but $g^4=g\notin H$.

lhf
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