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Group Homomorphism


A group homomorphism is a map f:G->H between two groups such that the group operation is preserved:f(g_1g_2)=f(g_1)f(g_2) for all g_1,g_2 in G, where the product on the left-hand side is in G and on the right-hand side in H.

As a result, a group homomorphism maps the identity element in G to the identity element in H: f(e_G)=e_H.

Note that a homomorphism must preserve the inverse map because f(g)f(g^(-1))=f(gg^(-1))=f(e_G)=e_H, so f(g)^(-1)=f(g^(-1)).

In particular, the image of G is a subgroup of H and the group kernel, i.e., f^(-1)(e_H) is a subgroup of G. The kernel is actually a normal subgroup, as is the preimage of any normal subgroup of H. Hence, any (nontrivial) homomorphism from a simple group must be injective.


See also

Homomorphism, Group, Group Representation, Normal Subgroup

Portions of this entry contributed by Todd Rowland

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References

Bronshtein, I. N. and Semendyayev, K. A. Handbook of Mathematics, 3rd ed. New York: Springer-Verlag, 1997.Scott, W. R. Group Theory. New York: Dover, 1987.

Referenced on Wolfram|Alpha

Group Homomorphism

Cite this as:

Rowland, Todd and Weisstein, Eric W. "Group Homomorphism." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GroupHomomorphism.html

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