With this noncommutativity in mind,
in general |
Of course, if + is commutative
then |
Even when + is not commutative, |
These are called the normal subgroups of G. |
If H is a normal subgroup of G, we can define an operation of cosets: |
(a + H) + (b + H) = (a + b) + H |
With this operation, the cosets of H form a group, called the
quotient group |
For example, consider Z6/H1. | |||||||||||||||||||
We have seen the cosets are |
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The function |
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Because H1 is isomorphic to Z2, we can write this isomorphism as | |||||||||||||||||||
Z6/Z2 = Z3 |
Return to Some group theory.