Notes on Cambridge Maths

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Subgroup

Let be a Group and .
We say that is a subgroup if is a Group.
We write

Proposition

Let be a Group and nonempty.
Suppose for any we have .
Then .

Proof

Firstly, is nonempty so let
Then so
Furthermore, for any we have
so
Finally, is a well defined function
as for any we have thus
so
Associativity is then inherited.

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