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coset (Definition)

Let $ H$ be a subgroup of a group $ G$, and let $ a \in G$. The left coset of $ a$ with respect to $ H$ in $ G$ is defined to be the set

$\displaystyle aH := \{ ah \mid h \in H \}. $
The right coset of $ a$ with respect to $ H$ in $ G$ is defined to be the set
$\displaystyle 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 index of $ H$ in $ G$, denoted $ [G:H]$, is the cardinality of the set $ G/H$ of left cosets of $ H$ in $ G$.



"coset" is owned by djao. [ full author list (2) ]
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Also defines:  index, left coset, right coset
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Cross-references: cardinality, relation, equivalence relation, partitions, collection, disjoint, group, subgroup
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This is version 5 of coset, born on 2002-01-05, modified 2002-11-04.
Object id is 1306, canonical name is Coset.
Accessed 20222 times total.

Classification:
AMS MSC20A05 (Group theory and generalizations :: Foundations :: Axiomatics and elementary properties)

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