Coset 2002-01-05 02:06:38 2002-11-04 09:06:16 Definition index left coset right coset \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{xypic} Let $H$ be a subgroup of a group $G$, and let $a \in G$. The {\em left coset} of $a$ with respect to $H$ in $G$ is defined to be the set $$ aH := \{ ah \mid h \in H \}. $$ The {\em right coset} of $a$ with respect to $H$ in $G$ is defined to be the set $$ Ha := \{ ha \mid h \in H \}. $$ Two left cosets $aH$ and $bH$ of $H$ in $G$ are either identical or disjoint. Indeed, if $c \in aH \cap bH$, then $c = ah_1$ and $c = bh_2$ for some $h_1,h_2 \in H$, whence $b^{-1} a = h_2 h_1^{-1} \in H$. But then, given any $ah \in aH$, we have $ah = (bb^{-1})ah = b(b^{-1}a) h \in bH$, so $aH \subset bH$, and similarly $bH \subset aH$. Therefore $aH = bH$. Similarly, any two right cosets $Ha$ and $Hb$ of $H$ in $G$ are either identical or disjoint. Accordingly, the collection of left cosets (or right cosets) partitions the group $G$; the corresponding equivalence relation for left cosets can be described succintly by the relation $a \sim b$ if $a^{-1} b \in H$, and for right cosets by $a \sim b$ if $ab^{-1} \in H$. The {\em index} of $H$ in $G$, denoted $[G:H]$, is the cardinality of the set $G/H$ of left cosets of $H$ in $G$.