Tour:Introduction seven (beginners)

This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)

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We completed part six by understanding normal subgroups as the kernels of homomorphisms, and showing that a surjective homomorphism is completely determined by its kernel. We now take another look at homomorphisms, leading up to an alternate description of normality. We'll use this to understand some metaproperties of normality, and then study some more isomorphism theorems.

We'll see the following definitions:

• Endomorphism of a group
• Automorphism of a group
• Automorphism group
• Inner automorphism
• Group acts as automorphisms by conjugation
• Normal subgroup (revisit)
• Normality is strongly intersection-closed
• Normality is strongly join-closed
• Normality satisfies intermediate subgroup condition
• Normality satisfies transfer condition
• Normality satisfies inverse image condition
• Second isomorphism theorem
• Third isomorphism theorem
• Fourth isomorphism theorem