Isoclinic groups have same derived length

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Statement

Suppose G_1 and G_2 are isoclinic groups. Then, the following are true:

  • G_1 is a solvable group if and only if G_2 is a solvable group.
  • If G_1 and G_2 are both solvable and nontrivial, then they have the same derived length. If either of them is trivial, the other may be nontrivial but must still be abelian (in which case we have derived lengths of zero and one).

Related facts

Proof

Given: Isoclinic groups G_1 and G_2.

To prove: G_1 is solvable if and only if G_2 is, and if so, they have the same derived length if both are nontrivial. If either is trivial, the other may be nontrivial but must be abelian.

Step no. Assertion/construction Facts used Previous steps used
1 G_1 is solvable if and only if its derived subgroup is solvable, and if so, the derived length of G_1 is one more than the derived length of its derived subgroup (unless G_1 is trivial). definition of solvable group, via the derived series. --
2 G_2 is solvable if and only if its derived subgroup is solvable, and if so, the derived length of G_2 is one more than the derived length of its derived subgroup (unless G_2 is trivial). definition of solvable group, via the upper central series. --
3 The derived subgroup of G_1 is isomorphic to the derived subgroup of G_2. definition of isoclinism G_1 is isoclinic to G_2.
4 G_1 is solvable if and only if G_2 is solvable, and they have the same derived length unless one of them is trivial. Steps (1)-(3)
5 If either group is trivial, the derived subgroup of both must be trivial, so both must be abelian. Step (3).