Group Extension

An extension of a group H by a group N is a group G with a normal subgroup M such that M=N and G/M=H. This information can be encoded into a short exact sequence of groups


where alpha:N->G is injective and beta:G->H is surjective.

It should be noted that some authors reverse the roles and say that H is an extension of N (Spanier 1994, Mac Lane and Birkhoff 1993).

Given groups H and N there are (often) many extensions of H by N. Examples include the direct product of H and N and a semidirect product of H and N. A function tau:H->G such that betatau is the identity function on H is called a transversal function. A group extension is said to be split if there is a transversal function which is a homomorphism. A group extension is split iff it is a semidirect product.

The study of group extensions has connections with group cohomology.

See also

Cohomology, Direct Product, Group, Normal Subgroup

This entry contributed by James Woodward

Explore with Wolfram|Alpha


Mac Lane, S. and Birkhoff, G. Algebra, 3rd ed. New York: Chelsea, 1993.Robinson, J. S. A Course in the Theory of Groups. New York: Springer-Verlag, pp. 310-313, 1998.Spanier, E. H. Algebraic Topology. New York: Springer-Verlag, 1994.

Referenced on Wolfram|Alpha

Group Extension

Cite this as:

Woodward, James. "Group Extension." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

Subject classifications