RankAb

rankAb is a group-theoretic invariant that measures the free abelian rank of a group's abelianization. For a group G, its abelianization is Gn = G/[G,G], where [G,G] is the commutator subgroup. rankAb(G) is the rank of Gn as an abelian group, i.e., the maximal integer r such that Z^r embeds into Gn. Equivalently, rankAb(G) equals the torsion-free rank of Gn, and for finitely generated groups it equals the dimension of Gn ⊗ Q as a vector space over Q. If Gn is finite, rankAb(G) = 0.

Basic properties include that rankAb is invariant under group isomorphism and is monotone under surjective homomorphisms: if φ: G → H is onto, then rankAb(G) ≥ rankAb(H). For common classes, free groups F_n have rankAb(F_n) = n, since F_n abelianizes to Z^n. Finite groups have rankAb = 0 because their abelianizations are finite. If G is abelian, rankAb(G) equals the rank of G itself.

Computation often proceeds from a presentation. If G = ⟨x1, ..., xm | R⟩, then Gn is presented by ⟨x1, ..., xm | commutativity, R⟩, and rankAb(G) equals m − rank(M), where M is the relation matrix obtained from the abelianized relations. Smith normal form provides a practical method to determine this rank.

Relation to topology is standard: for a space X with π1(X) ≅ G, rankAb(G) equals the first Betti number b1(X) = rank H1(X; Z). Examples include G = Z^n (rankAb = n) and G = S_n (n ≥ 3) which has rankAb = 0.

Applications of rankAb include distinguishing groups with the same finite quotient structure and informing questions about generating sets and abelian quotients.