Isomorphic inner automorphism groups and isomorphic derived subgroups not implies isoclinic

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Statement

It is possible to have two groups G and H such that:

  1. Their inner automorphism groups are isomorphic, i.e., \operatorname{Inn}(G) \cong \operatorname{Inn}(H).
  2. Their derived subgroups are isomorphic, i.e., G' \cong H'.
  3. G and H are not isoclinic groups

The key is that despite the inner automorphism groups being isomorphic and the derived subgroups being isomorphic, we cannot choose isomorphisms that are compatible with the commutator mapping, which is what would be required for an isoclinism.

Proof

Example of groups of order 64

The smallest order examples for 2-groups occurs for groups of order 64. There are in fact two examples at this order where in each example there are three different equivalence classes up to isoclinism that share an inner automorphism group and derived subgroup.

First example

The inner automorphism group for this example is direct product of D8 and Z2 and the derived subgroup is direct product of Z4 and Z2.

The three equivalence classes up to isoclinism can be described as follows:

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Second example

The inner automorphism group is elementary abelian group:E16 and the derived subgroup is Klein four-group.

The three equivalence classes up to isoclinism can be described as follows:

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