Magma is a set with a binary operation. (and that's all)
f[A ⊕ B] = f[A] ⊕ f[B] f[s ⊙ A] = s ⊙ f[A]
A group (G, *) is a set G with a binary operation *.
Order of a group G, is the number of elements in it. It can be finit, or infinite.
The Order of a element x, is the number n, such that x^n == e.
A set H is a subgroup of a group G if it is a subset of G and a group using the operation defined on G. In other words, H is a subgroup of (G, *) if the restriction of * to H is a group operation on H.
If G is a finite group, then so is H. Further, the order of H divides the order of G (Lagrange's Theorem).
direct product of a group
If (G,*) and (H,•) are groups, then the set G×H together with the operation (g1,h1)(g2,h2) = (g1*g2,h1•h2) is a group.
The diff between ismorphism and homomorphism is that the former is bijection.
Given two groups (G, *) and (H, @), a group isomorphism from (G, *) to (H, @) is a bijective group homomorphism from G to H. Spelled out, this means that a group isomorphism is a bijective function f : G → H such that for all u and v in G it holds that
f (u * v) = f (u) @ f (v)
The fundamental theorem of finite abelian groups states that every finite abelian group can be expressed as the direct sum of cyclic subgroups of prime-power order. Lagrange's theorem (group theory)
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