A homomorphism is a function from one AlgebraicGroup into another which preserves the group structure.  Formally, if ''G'' and ''H'' are groups with operations * and #, respectively, and ''f'' is a function from ''G'' to ''H'', then ''f'' is a homomorphism if

	 :	''a'' * ''b'' = ''c''  implies  ''f''(''a'') # ''f''(''b'') = ''f''(''c'')

for all ''a'', ''b'', and ''c'' in ''G''.

A homomorphism maps the identity element of ''G'' onto the identity element of ''H''.  If ''h'' is the image of ''g'' under a homomorphism, then the inverse of ''h'' is the image of the inverse of ''g''.

The definition above is not the usual one, but it is equivalent to it.  The usual definition is that ''f''(''a''*''b'') = ''f''(''a'') # ''f''(''b'') for all ''a'' and ''b'' in ''G''.

If a homomorphism is one-to-one and onto, then its inverse is also a homomorphism, and it is called an ''isomorphism''.  Two groups are considered to be the same if there is an isomorphism between them.

If ''H'' is a group with identity element ''e'', and ''f'' is a homomorphism from ''G'' to ''H'', then the set of all elements  ''g'' in ''G'' such that ''f''(''g'') = ''e'' is called the ''kernel'' of the homomorphism.  The kernel of a homomorphism is always a NormalSubgroup of ''G''.

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CategoryMath