In abstract algebra, 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).
Table of contents |
2 Consequences 3 Automorphisms |
Examples
The group of all real numbers with addition, (R,+), is isomorphic to the group of all positive real numbers with multiplication (R^{+},×) via the isomorphism
- f(x) = exp(x)
The group Z of integers (with addition) is a subgroup of R, and the factor group R/Z is isomorphic to the group S^{1} of complex numbers of absolute value 1 (with multiplication); an isomorphism is given by
- f(x + Z) = exp(2πxi)
The Klein four-group is isomorphic to the direct product of two copies of Z/2Z (see modular arithmetic).
Consequences
that it will map inverses to inverses,- f(u^{-1}) = f(u)^{-1}
The relation "being isomorphic" satisfies all the axioms of an equivalence relation. If f is an isomorphism between G and H, then everything that is true about G can be translated via f into a true statement about H, and vice versa.