Stem group has the minimum order among all groups isoclinic to it

Statement

Suppose $G$ is a stem group (i.e., $Z(G)\leq [G,G]$) and $H$ is a group that is isoclinic to $G$. Then, the order of $G$ is less than or equal to the order of $H$. This statement holds both in the case of finite groups (where the order is just a finite number) and infinite groups (if we use infinite cardinals).

In the finite case, it is further true that the order of $G$ divides the order of $H$.

Related facts

• Every group is isoclinic to a stem group

References

Journal references

• The classification of prime-power groups by Philip Hall, Volume 69, (Year 1937): ^{Official linkMore info}: Page 135 (Page 6 within the paper) alludes to this fact in the finite context, with a proof of sorts around the statement.