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Left cosets partition a group

9 bytes added, 00:04, 4 July 2011
Proof in form (2)
'''To prove''': For any <math>a \in G</math>, <math>a \sim a</math>.
'''Proof''': Clearly <math>e \in H</math> (since <math>H</math> is a subgroup). Hence, for any <math>a \in G</math>, <math>a = ae</math>, so <math>\! a \sim a</math>: <math>a</math> is in its own left coset.
===Symmetry===
'''To prove''': For any <math>a,b \in G</math> such that <math>\! a \sim b</math>, we have <math>\! b \sim a</math>.
'''Proof''': If <math>a = bh</math>, for some <math>h \in H</math>, then <math>b = ah^{-1}</math>. Since <math>h \in H</math> and <math>H</math> is a subgroup, <math>h^{-1} \in H</math>. Thus, if <math>a</math> is in the left coset of <math>b</math>, then <math>b</math> is in the left coset of <math>a</math>. In symbols, <math>a \sim b \implies b \sim a</math>.
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