Rank of a group

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

If G is a finitely generated group, then the rank of G is a non-negative integer. The notion of rank of a group is a group-theoretic analog of the notion of dimension of a vector space. Indeed, for p-groups, the rank of the group P is the dimension of the vector space P/Φ(P), where Φ(P) is the Frattini 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 as affine groups. To distinguish these different definitions, one sometimes calls this rank the subgroup rank. Explicitly, the subgroup rank of a group G 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.

Known facts and examples

• For a nontrivial group G, we have rank(G) = 1 if and only if G is a cyclic group. The trivial group T has rank(T) = 0, since the minimal generating set of T is the empty set.
• For the free abelian group ${\displaystyle {\mathbb{Z}}^n}$, we have ${\displaystyle \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}({\mathbb{Z}}^n)=n.}$
• If X is a set and G = F(X) is the free group with free basis X then rank(G) = |X|.
• If a group H is a homomorphic image (or a quotient group) of a group G then rank(H) ≤ rank(G).
• If G is a finite non-abelian simple group (e.g. G = A_n, the alternating group, for n > 4) then rank(G) = 2. This fact is a consequence of the Classification of finite simple groups.
• If G is a finitely generated group and Φ(G) ≤ G is the Frattini subgroup of G (which is always normal in G so that the quotient group G/Φ(G) is defined) then rank(G) = rank(G/Φ(G)).
• If G is the fundamental group of a closed (that is compact and without boundary) connected 3-manifold M then rank(G)≤g(M), where g(M) is the Heegaard genus of M.
• If H,K ≤ F(X) are finitely generated subgroups of a free group F(X) such that the intersection ${\displaystyle L=H\cap K}$ is nontrivial, then L is finitely generated and

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

This result is due to Hanna Neumann. The Hanna Neumann conjecture states that in fact one always has rank(L) − 1 ≤ (rank(K) − 1)(rank(H) − 1). The Hanna Neumann conjecture has recently been solved by Igor Mineyev and announced independently by Joel Friedman.

• According to the classic Grushko theorem, rank behaves additively with respect to taking free products, that is, for any groups A and B we have

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

• If ${\displaystyle G=⟨x_1,\dots ,x_n|r=1⟩}$ is a one-relator group such that r is not a primitive element in the free group F(x₁,..., x_n), that is, r does not belong to a free basis of F(x₁,..., x_n), then rank(G) = n.
• 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. If G is a crystallographic point group, then rank(G) is up to 3. If G is a wallpaper group, then rank(G) = 2 to 4. The only wallpaper-group type of rank 4 is p2mm. If G is a 3-dimensional space group, then rank(G) = 2 to 6. The only space-group type of rank 6 is Pmmm.