Some Theorems
Every group G has a presentation. To see this, consider the free group FG on G. By the universal property of free groups, there exists a unique group homomorphism φ : FG → G whose restriction to G is the identity map. Let K be the kernel of this homomorphism. Then K is normal in FG, therefore is equal to its normal closure, so <G|K> = FG/K. Since the identity map is surjective, φ is also surjective, so by the First Isomorphism Theorem, <G|K> = G. Note that this presentation may be highly inefficient if both G and K are much larger than necessary.
Every finite group has a finite presentation: one may take the elements of the group for generators and the Cayley table for relations.
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