# Isomorphic iff potentially conjugate in finite

## Contents

## Statement

### With one isomorphism

Suppose is a finite group and is an isomorphism between two subgroups of . Then, there exists a finite group containing and an element such that for any .

### With multiple isomorphisms

Suppose is a finite group and is a collection of isomorphisms between subgroups of , with . Then, there exists a finite group containing and elements such that for any .

## Related facts

- Isomorphic iff potentially conjugate: The general version for infinite groups. Note that the proof outlined in the finite case does not work directly for infinite groups, because the finite groups version uses the fact that isomorphic subgroups have the same index.
- Equivalence of definitions of order statistics-equivalent finite groups: Two finite groups that have the same order statistics can be embedded in a symmetric group in such a way that for every conjugacy class of the symmetric group, the intersection of both groups with the conjugacy class has the same size. Conversely, if two finite groups can be embedded in any big group with this property, they have the same order statistics.

- Inner automorphism to automorphism is right tight for normality
- Every injective endomorphism arises as the restriction of an inner automorphism
- Normal upper-hook fully normalized implies characteristic
- Left transiter of normal is characteristic

## Examples

In our examples, the finite group that we construct is much smaller than the needed for the general construction.

and | Isomorphism class of | and embedding of in | Embeddings of in | |
---|---|---|---|---|

Klein four-group | any two subgroups of order two (Z2 in V4) | cyclic group:Z2 | dihedral group:D8, V4 in D8 | non-normal subgroups of dihedral group:D8 |

Dihedral group:D8 | center of dihedral group:D8, one of the non-normal subgroups of dihedral group:D8 | cyclic group:Z2 | symmetric group:S4, D8 in S4 | |

Dihedral group:D8 | the two Klein four-subgroups of dihedral group:D8 | Klein four-group | dihedral group:D16, D8 in D16 | V4 in D16 |

## Proof

### With one isomorphism

Let be the symmetric group on the set with viewed as a subgroup of via the left multiplication action (this is the embedding given by Cayley's theorem). Under this embedding, the isomorphism lifts to an inner automorphism given by conjugation by a permutation constructed as follows.

The restriction of to is . Further, pick right coset representatives for in such that the identity element is the coset representative of the subgroups themselves. Now, define as a bijection between these sets of coset representatives such that sends the identity element to the identity element. Finally, for any element of the form , with a coset representative of and , define .

We verify that this works. Pick and an element . Write , with a coset representative of and . Let denote the left multiplication by . We want to show that .

The left side is . For the right side, note that , so , with and a right coset representative of . Thus, the right side is . Since and is a right coset representative, we get that . This equals the left side, so the proof is completed.

### With multiple isomorphisms

Since is finite, so is the set of isomorphisms. Without loss of generality, . The rough idea is that each time we take the symmetric group on the set, we can convert one isomorphism to an inner automorphism. Thus, repeating the process times yields a finite group in which all the isomorphisms are realized as inner automorphisms.

The new finite group has order , where is the factorial function. This grows very rapidly with for . For instance, is which is a number with several thousand digits.