Group automorphism ( $α : G \sim G$ )

in the figure under endomorphism. The general criterion under endomorphism of a finite cyclic group says exactly when such a map is reversible: its multiplier must be coprime to the size of the finite cyclic group.

A permutation need not be a group automorphism, even if it fixes the identity element. Let

π

exchange

1

and

2

C_{6}

and fix

0, 3, 4, 5

. It is a bijection, and

π (0) = 0

, but

π (1 + 1) = π (2) = 1, π (1) + π (1) = 2 + 2 = 4 (mod 6) .

(1)

Thus it fails to preserve addition and is not even a group homomorphism. Reversibility alone does not preserve the group operation.

Figure 1.
Addition before or after relabeling: agreement and disagreement modulo six
. Each panel follows the same input pair $(1, 1)$ along two routes: add then relabel, or relabel each entry then add. All addition is modulo six. The routes agree for the group automorphism $f_{5}$ ; for the permutation exchanging $1$ and $2$ , this single disagreement disproves preservation of addition. **Source**. **Details**.

A group automorphism carries each cyclic subgroup onto the cyclic subgroup generated by the image of its defining element, preserving the order of a group element. In particular, it carries a generator of a group to another generator of a group. For

C_{6}

, the existing calculation of generators of a finite cyclic group leaves only the images

1

and

5

for

1

, so its automorphism group consists exactly of the identity function and

f_{5}

. This turns the problem of compatible relabeling into a much smaller choice than enumerating arbitrary permutations.

Table of contents
- Characteristic subgroup Group automorphism
  - Characteristically simple group Characteristic subgroup

Characteristic subgroup

 1  0

A subgroup is characteristic when every automorphism of the ambient group maps it to itself. A uniquely determined subgroup of a given order or structural type is characteristic.