# Rank of a group

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For the dimension of the Cartan subgroup, see Rank of a Lie group

In the mathematical subject of group theory, the rank of a groupG, denoted rank(G), can refer to the smallest cardinality of a generating set for G, that is Mathematics includes the study of such topics as quantity, structure, space, and change. In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.

In mathematics, the cardinality of a set is a measure of the "number of elements of the set". For example, the set contains 3 elements, and therefore has a cardinality of 3. There are two approaches to cardinality – one which compares sets directly using bijections and injections, and another which uses cardinal numbers. The cardinality of a set is also called its size, when no confusion with other notions of size is possible.

## Contents

$\operatorname {rank} (G)=\min\{|X|:X\subseteq G,\langle X\rangle =G\}.$ If G is a finitely generated group, then the rank of G is a nonnegative 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. In algebra, a finitely generated group is a group G that has some finite generating set S so that every element of G can be written as the combination of finitely many elements of the finite set S and of inverses of such elements. In mathematical group theory, given a prime number p, a p-group is a group in which each element has a power of p as its order. That is, for each element g of a p-group, there exists a nonnegative integer n such that the product of pn copies of g, and not fewer, is equal to the identity element. The orders of different elements may be different powers of p. In mathematics, the Frattini subgroup Φ(G) of a group G is the intersection of all maximal subgroups of G. For the case that G has no maximal subgroups, for example the trivial group e or the Prüfer group, it is defined by Φ(G) = G. It is analogous to the Jacobson radical in the theory of rings, and intuitively can be thought of as the subgroup of "small elements". It is named after Giovanni Frattini, who defined the concept in a paper published in 1885.

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:

In mathematics, the affine group or general affine group of any affine space over a field K is the group of all invertible affine transformations from the space into itself.

$\operatorname {sr} (G)=\max _{H\leq G}\min\{|X|:X\subseteq H,\langle X\rangle =H\}.$ 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 a free abelian group $\mathbb {Z} ^{n}$ we have ${\rm {rank}}(\mathbb {Z} ^{n})=n.$ • 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 = An, 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,KF(X) are finitely generated subgroups of a free group F(X) such that the intersection $L=H\cap K$ is nontrivial, then L is finitely generated and In group theory, a branch of abstract algebra, a cyclic group or monogenous group is a group that is generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, and it contains an element g such that every other element of the group may be obtained by repeatedly applying the group operation to g or its inverse. Each element can be written as a power of g in multiplicative notation, or as a multiple of g in additive notation. This element g is called a generator of the group. In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced. Many possible properties of sets are vacuously true for the empty set.

In abstract algebra, a free abelian group or free Z-module is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis is a subset such that every element of the group can be found by adding or subtracting basis elements, and such that every element's expression as a linear combination of basis elements is unique. For instance, the integers under addition form a free abelian group with basis {1}. Addition of integers is commutative, associative, and has subtraction as its inverse operation, each integer is the sum or difference of some number of copies of the number 1, and each integer has a unique representation as an integer multiple of the number 1.

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

In the mathematical subject of group theory, the Grushko theorem or the Grushko-Neumann theorem is a theorem stating that the rank of a free product of two groups is equal to the sum of the ranks of the two free factors. The theorem was first obtained in a 1940 article of Grushko and then, independently, in a 1943 article of Neumann.

In mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new group GH. The result contains both G and H as subgroups, is generated by the elements of these subgroups, and is the “universal” group having these properties, in the sense that any two homomorphisms from G and H into a group K factor uniquely through an homomorphism from GH to K. Unless one of the groups G and H is trivial, the free product is always infinite. The construction of a free product is similar in spirit to the construction of a free group.

rank(A$\ast$ B) = rank(A) + rank(B).
• If $G=\langle x_{1},\dots ,x_{n}|r=1\rangle$ is a one-relator group such that r is not a primitive element in the free group F(x1,..., xn), that is, r does not belong to a free basis of F(x1,..., xn), then rank(G) = n.  

In algebra, a primitive element of a co-algebra C is an element x that satisfies

## The rank problem

There is an algorithmic problem studied in group theory, known as the rank problem. The problem asks, for a particular class of finitely 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 of a finitely generated group G can be equivalently defined as the smallest cardinality of a set X such that there exists an onto homomorphism F(X) → G, where F(X) is the free group with free basis X. There is a dual notion of co-rank of a finitely generated group G defined as the largest cardinality of X such that there exists an onto homomorphism GF(X). Unlike rank, co-rank is always algorithmically computable for finitely presented groups,  using the algorithm of Makanin and Razborov for solving systems of equations in free groups.   The notion of co-rank is related to the notion of a cut number for 3-manifolds. 

If p is a prime number, then the p-rank of G is the largest rank of an elementary abelian p-subgroup.  The sectionalp-rank is the largest rank of an elementary abelian p-section (quotient of a subgroup).

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