Transformation degree equals permutation degree for groups

Exact statement

For every finite group (G), the minimum (m) admitting an embedding $G\hookrightarrow T_m$ equals the minimum faithful permutation degree μ(G).

StatusProved; standard group-action fact

External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

This identifies the group question as the literal group restriction of the transformation-monoid question.

Hypotheses and scope

G is finite.

Proof or evidence

Under a unital embedding, every image element has a two-sided inverse and is therefore a permutation. Under a non-unital embedding, restriction to the image of the identity gives a faithful permutation action of no larger degree. The reverse inequality follows from S_m⊆T_m.

Equality of the two degrees →

Verification notes

The proof was checked under both unital and non-unital conventions.

Limitations

This equality does not decide how hard it is to compute the common degree from generators.

Open work

Use standard minimum-faithful-action terminology in later group-specific work.