Class-preserving implies IA

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This article gives the statement and possibly, proof, of an implication relation between two automorphism properties. That is, it states that every automorphism satisfying the first automorphism property (i.e., class-preserving automorphism) must also satisfy the second automorphism property (i.e., IA-automorphism)
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Property-theoretic statement

The automorphism property of being a Class-preserving automorphism (?) is stronger than the automorphism property of being an IA-automorphism (?).

Definitions used

Class automorphism

Further information: Class-preserving automorphism

An automorphism of a group is termed a class automorphism if it sends every element to an element in its conjugacy class; in other words, it preserves conjugacy classes.


Further information: IA-automorphism

An automorphism of a group is termed an IA-automorphism if it induces the identity map on the Abelianization of the group. In other words, it sends every coset of the commutator subgroup, to itself.