Groupprops, The Group Properties Wiki (pre-alpha)

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From Groupprops

This article gives a proof/explanation of the equivalence of multiple definitions for the term normal subgroup
View a complete list of pages giving proofs of equivalence of definitions

Contents 1 Statement 2 Facts used 3 Proof 3.1 Conceptual proof using function restriction expressions

Statement

The following are equivalent for a subgroup of a group:

• The subgroup is a normal subgroup: it is invariant under all inner automorphisms of the whole group.
• The subgroup is an extensible automorphism-invariant subgroup: it is invariant under all extensible automorphisms of the whole group.

Facts used

1. Inner implies extensible
2. Extensible implies normal

Proof

Conceptual proof using function restriction expressions

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