TITLE: Showing equivalence of the cosets
QUESTION [1 upvotes]: Show that if H is a subgroup of index 2 in a finite group G, then every left coset of H is also right coset of H.
Thanks for your helps..

REPLY [1 votes]: $[G:H] = 2 \Longrightarrow G$ is being partitioned into $2$ disjoint cosets.  If $g \in H$, then $gH = Hg = H$.  Since there are only two partitions, then $g \notin H \Rightarrow gH = Hg = G$ \ $H$.  
Therefore, for all $g \in G$, $gH = Hg$.
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