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created: 2021-12-17 18:48:03
modified: 2022-01-10 04:13:04
Every group is isomorphic to a subgroup of the symmetric group acting on .
Proof
For any let is define . Since , i.e. it exists, every represents a permutation of . We can write then , which enables us to define the function with .
- is injective since there is a one-to-one correspondence between and (via )
- is a homomorphism:
- The homomorphism theorem tells (or it is east to show directly as well) that is also a group, more precisely a subgroup of
Since , we are done. QED