In themathematical subject ofgroup theory, therank of a groupG, denoted rank(G), can refer to the smallestcardinality of agenerating set forG, that is

${\displaystyle \mathrm{rank}(G)=min\{|X|:X\subseteq G,⟨X⟩=G\}.}$

IfG is afinitely generated group, then the rank ofG is a non-negativeinteger. The notion of rank of a group is a group-theoretic analog of the notion ofdimension of a vector space. Indeed, forp-groups, the rank of the groupP is the dimension of the vector spaceP/Φ(P), where Φ(P) is theFrattini subgroup.

The rank of a group is also often defined in such a way as to ensure subgroups have rank less than or equal to the whole group, which is automatically the case for dimensions of vector spaces, but not for groups such asaffine groups. To distinguish these different definitions, one sometimes calls this rank thesubgroup rank. Explicitly, the subgroup rank of a groupG is the maximum of the ranks of its subgroups:

${\displaystyle \mathrm{sr}(G)=\underset{H\le G}{max}min\{|X|:X\subseteq H,⟨X⟩=H\}.}$

Sometimes the subgroup rank is restricted to abelian subgroups.

• For a nontrivial groupG, we have rank(G) = 1 if and only ifG is acyclic group. The trivial groupT has rank(T) = 0, since the minimal generating set ofT is theempty set.
• For thefree abelian group${\displaystyle {\mathbb{Z}}^n}$, we have${\displaystyle \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}({\mathbb{Z}}^n)=n.}$
• IfX is a set andG =F(X) is thefree group with free basisX then rank(G) = |X|.
• If a groupH is ahomomorphic image (or aquotient group) of a groupG then rank(H) ≤ rank(G).
• IfG is a finite non-abeliansimple group (e.g.G = A_n, thealternating group, forn > 4) then rank(G) = 2. This fact is a consequence of theClassification of finite simple groups.
• IfG is a finitely generated group and Φ(G) ≤G is theFrattini subgroup ofG (which is always normal inG so that the quotient groupG/Φ(G) is defined) then rank(G) = rank(G/Φ(G)).^[1]
• IfG is thefundamental group of a closed (that iscompact and without boundary) connected3-manifoldM then rank(G)≤g(M), whereg(M) is theHeegaard genus ofM.^[2]
• IfH,K ≤F(X) arefinitely generated subgroups of afree groupF(X) such that the intersection${\displaystyle L=H\cap K}$ is nontrivial, thenL is finitely generated and

rank(L) − 1 ≤ 2(rank(K) − 1)(rank(H) − 1).

This result is due toHanna Neumann.^[3][4] TheHanna Neumann conjecture states that in fact one always has rank(L) − 1 ≤ (rank(K) − 1)(rank(H) − 1). TheHanna Neumann conjecture has recently been solved by Igor Mineyev^[5] and announced independently by Joel Friedman.^[6]

• According to the classicGrushko theorem, rank behaves additively with respect to takingfree products, that is, for any groupsA andB we have

rank(A${\displaystyle \ast }$B) = rank(A) + rank(B).

• If${\displaystyle G=⟨x_1,\dots ,x_n|r=1⟩}$ is aone-relator group such thatr is not aprimitive element in the free groupF(x₁,...,x_n), that is,r does not belong to a free basis ofF(x₁,...,x_n), then rank(G) = n.^[7][8]
• The rank of a symmetry group is closely related to the complexity of the object (a molecule, a crystal structure) being under the action of the group. IfG is acrystallographic point group, then rank(G) is up to 3.^[9] IfG is awallpaper group, then rank(G) = 2 to 4. The only wallpaper-group type of rank 4 isp2mm.^[10] IfG is a 3-dimensionalspace group, then rank(G) = 2 to 6. The only space-group type of rank 6 isPmmm.^[11]

There is an algorithmic problem studied ingroup theory, known as therank problem. The problem asks, for a particular class offinitely presented groups if there exists an algorithm that, given a finite presentation of a group from the class, computes the rank of that group. The rank problem is one of the harder algorithmic problems studied in group theory and relatively little is known about it. Known results include:

• The rank problem is algorithmically undecidable for the class of allfinitely presented groups. Indeed, by a classicalresult of Adian–Rabin, there is no algorithm to decide if a finitely presented group is trivial, so even the question of whether rank(G)=0 is undecidable for finitely presented groups.^[12][13]
• The rank problem is decidable for finite groups and for finitely generatedabelian groups.
• The rank problem is decidable for finitely generatednilpotent groups. The reason is that for such a groupG, theFrattini subgroup ofG contains thecommutator subgroup ofG and hence the rank ofG is equal to the rank of theabelianization ofG.^[14]
• The rank problem is undecidable forword hyperbolic groups.^[15]
• The rank problem is decidable for torsion-freeKleinian groups.^[16]
• The rank problem is open for finitely generated virtually abelian groups (that is containing an abelian subgroup of finiteindex), for virtually free groups, and for3-manifold groups.

The rank of afinitely generated groupG can be equivalently defined as the smallest cardinality of a setX such that there exists an ontohomomorphismF(X) →G, whereF(X) is thefree group with free basisX. There is a dual notion ofco-rank of afinitely generated groupG defined as thelargestcardinality ofX such that there exists an ontohomomorphismG →F(X). Unlike rank, co-rank is always algorithmically computable forfinitely presented groups,^[17] using the algorithm ofMakanin andRazborov for solving systems of equations in free groups.^[18][19]The notion of co-rank is related to the notion of acut number for3-manifolds.^[20]

Ifp is aprime number, then thep-rank ofG is the largest rank of anelementary abelianp-subgroup.^[21] Thesectionalp-rank is the largest rank of an elementary abelianp-section (quotient of a subgroup).

• Rank of an abelian group
• Prüfer rank
• Grushko theorem
• Free group
• Nielsen equivalence

• Group theory