# Thread: Commutator subgroup of a P-group.

1. ## Commutator subgroup of a P-group.

THE COMMUTATOR SUBGROUP OF A FINITE 2-GENERATED P-GROUP OF (NILPOTENCY) CLASS 2 IS CYCLIC

2. Originally Posted by arvie87

THE COMMUTATOR SUBGROUP OF A FINITE 2-GENERATED P-GROUP OF (NILPOTENCY) CLASS 2 IS CYCLIC

This is theorem 5.2.18 in Robinson's "A Course in the Theory of Groups" (2nd ed. 1996, page 137) .

Tonio

3. ## order of the commutator subgroup

maybe someone know the following theorem and knows how to prove or any book, where I can find it:

Let G be a p-group. The indices of centralizers are bounded by p for every x in G. Then, the order of the commutator subgroup is bounded by p:
|G| = p^n, |G:C_G(x)| <= p for all x in G => |G'| <= p

4. Originally Posted by nelig100
maybe someone know the following theorem and knows how to prove or any book, where I can find it:

Let G be a p-group. The indices of centralizers are bounded by p for every x in G. Then, the order of the commutator subgroup is bounded by p:
|G| = p^n, |G:C_G(x)| <= p for all x in G => |G'| <= p

Do not "kidnap" other poster's thread: begin a new one with your question in it.

Tonio