Group homomorphism
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 hand side of the equation is that of G and on the right hand 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, though this may be confused as an index or a general subscript. A more recent trend is to write group homomorphisms on the right of their arguments, omitting brackets, so that h(x) becomes simply x h. This approach is especially prevalent in areas of group theory where automata play a role, since it accords better with the convention that automata read words from left to right.
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.
We define the kernel of h to be the set of elements in G which are mapped to the identity in H
and the image of h to be
The kernel and image of a homomorphism can be interpreted as measuring how close it is to being an isomorphism. The first isomorphism theorem states that the image of a group homomorphism, h(G) is isomorphic to the quotient group G/ker h.
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