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# First Group Isomorphism Theorem

The first group isomorphism theorem, also known as the fundamental homomorphism theorem, states that if is a group homomorphism, then and , where indicates that is a normal subgroup of , denotes the group kernel, and indicates that and are isomorphic groups.

A corollary states that if is a group homomorphism, then

1. is injective iff

2. , where denotes the group order of a group .

Second Group Isomorphism Theorem, Third Group Isomorphism Theorem, Fourth Group Isomorphism Theorem

This entry contributed by Nick Hutzler

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## References

Dummit, D. S. and Foote, R. M. Abstract Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, pp. 98-100, 1998.

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First Group Isomorphism Theorem

## Cite this as:

Hutzler, Nick. "First Group Isomorphism Theorem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/FirstGroupIsomorphismTheorem.html