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Concept
Group homomorphism
Summary
In mathematics, given two groups, (G, ∗) and (H, ·), a group homomorphism from (G, ∗) to (H, ·) is a function h : G → H such that for all u and v in G it holds that
where the group operation on the left side of the equation is that of G and on the right side that of H.
From this property, one can deduce that h maps the identity element eG of G to the identity element eH of H,
and it also maps inverses to inverses in the sense that
Hence one can say that h "is compatible with the group structure".
Older notations for the homomorphism h(x) may be xh or xh, though this may be confused as an index or a general subscript. In automata theory, sometimes homomorphisms are written to the right of their arguments without parentheses, so that h(x) becomes simply .
In areas of mathematics where one considers groups endowed with additional structure, a homomorphism sometimes means a map which respects not only the group structure (as above) but also the extra structure. For example, a homomorphism of topological groups is often required to be continuous.
The purpose of defining a group homomorphism is to create functions that preserve the algebraic structure. An equivalent definition of group homomorphism is: The function h : G → H is a group homomorphism if whenever
a ∗ b = c we have h(a) ⋅ h(b) = h(c).
In other words, the group H in some sense has a similar algebraic structure as G and the homomorphism h preserves that.
Monomorphism A group homomorphism that is injective (or, one-to-one); i.e., preserves distinctness.
Epimorphism A group homomorphism that is surjective (or, onto); i.e., reaches every point in the codomain.
Isomorphism A group homomorphism that is bijective; i.e., injective and surjective. Its inverse is also a group homomorphism. In this case, the groups G and H are called isomorphic; they differ only in the notation of their elements and are identical for all practical purposes.
Endomorphism A group homomorphism, h: G → G; the domain and codomain are the same. Also called an endomorphism of G.
Official source
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