Stem group
A group G is a stem group if its center lies inside its derived subgroup:
- Z(G) ≤ [G, G].
Two groups are isoclinic when they have the same commutator structure: there are isomorphisms G/Z(G) → H/Z(H) and [G, G] → [H, H] that agree with the commutator, so [x, y] on one side matches [x, y] on the other. Isoclinism sorts all finite groups into families, and the stem groups of a family are exactly its members of least order.
Why the smallest
Every group in a family has the same G/Z(G), so |G| = |G/Z(G)| · |Z(G)|, and |Z(G) ∩ [G, G]| is the same for all of them. Then
- |G| = |G/Z(G)| · |Z(G)| ≥ |G/Z(G)| · |Z(G) ∩ [G, G]|,
with equality exactly when Z(G) ≤ [G, G]. So being stem and being smallest in the family are the same condition, and every family has a smallest order that its stem groups share.
Key facts
- Containment is enough — Z(G) need not equal [G, G]. Both happen: in
D₄the two coincide, while this group of order 128 is stem with a center of order 4 sitting properly inside a derived subgroup of order 16. - Every centerless group is stem: Z(G) = {e} lies in every subgroup. So
S₃,A₅and every nonabelian simple group qualify. - Every perfect group is stem: [G, G] = G contains everything, the center included.
- Among abelian groups only the trivial group is stem — Z(G) = G but [G, G] = {e}. All abelian groups are isoclinic to one another, and the trivial group is their family's stem group.
- A family can hold several non-isomorphic stem groups, so "the" stem group of a family is not well defined. Its order is.
- Isoclinic groups of the same order can be non-isomorphic while sharing invariants that commutators control — the reason
D₄andQ₈have the same character table.
Examples
D₄andQ₈, both of order 8: Z = [G, G] = C₂ in each. They are isoclinic, and both are stem groups of that family.S₃, order 6 — centerless, so stem.- The trivial group, the stem group of the abelian family.
Non-examples
M₁₆, the modular group of order 16: Z(G) ≅ C₄ but [G, G] ≅ C₂, so the center is too big. It is isoclinic to D₄ and Q₈, and at twice their order it is not stem.- D₄ × C₂, order 16: the center grows to C₂² while the derived subgroup stays C₂. Same family again.
- Every nontrivial abelian group,
C₂upward.