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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 group***G*, 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.

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

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.

Sometimes the subgroup rank is restricted to abelian subgroups.

- 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 we have
- 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*, the alternating group, for_{n}*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*)).^{ [1] } - 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*.^{ [2] } - If
*H*,*K*≤*F*(*X*) are finitely generated subgroups of a free group*F*(*X*) such that the intersection 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.
^{ [3] }^{ [4] }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^{ [5] }and announced independently by Joel Friedman.^{ [6] }

- 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 *G* ∗ *H*. 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 *G* ∗ *H* 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**B*) = rank(*A*) + rank(*B*).

- If is a one-relator group such that
*r*is not a primitive element in the free group*F*(*x*_{1},...,*x*_{n}), that is,*r*does not belong to a free basis of*F*(*x*_{1},...,*x*_{n}), then rank(*G*) =*n*.^{ [7] }^{ [8] }

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

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 problem is algorithmically undecidable for the class of all finitely presented groups. Indeed, by a classical result 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.^{ [9] }^{ [10] } - The rank problem is decidable for finite groups and for finitely generated abelian groups.
- The rank problem is decidable for finitely generated nilpotent groups. The reason is that for such a group
*G*, the Frattini subgroup of*G*contains the commutator subgroup of*G*and hence the rank of*G*is equal to the rank of the abelianization of*G*.^{ [11] } - The rank problem is undecidable for word hyperbolic groups.
^{ [12] } - The rank problem is decidable for torsion-free Kleinian groups.
^{ [13] } - The rank problem is open for finitely generated virtually abelian groups (that is containing an abelian subgroup of finite index), for virtually free groups, and for 3-manifold groups.

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 *G* → *F*(*X*). Unlike rank, co-rank is always algorithmically computable for finitely presented groups,^{ [14] } using the algorithm of Makanin and Razborov for solving systems of equations in free groups.^{ [15] }^{ [16] } The notion of co-rank is related to the notion of a *cut number* for 3-manifolds.^{ [17] }

If *p* is a prime number, then the *p*-**rank** of *G* is the largest rank of an elementary abelian *p*-subgroup.^{ [18] } The **sectional***p*-**rank** is the largest rank of an elementary abelian *p*-section (quotient of a subgroup).

- ↑ D. J. S. Robinson.
*A course in the theory of groups*, 2nd edn, Graduate Texts in Mathematics 80 (Springer-Verlag, 1996). ISBN 0-387-94461-3 - ↑ Friedhelm Waldhausen.
*Some problems on 3-manifolds.*Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, pp. 313–322, Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978; ISBN 0-8218-1433-8 - ↑ Hanna Neumann.
*On the intersection of finitely generated free groups.*Publicationes Mathematicae Debrecen, vol. 4 (1956), 186–189. - ↑ Hanna Neumann.
*On the intersection of finitely generated free groups. Addendum.*Publicationes Mathematicae Debrecen, vol. 5 (1957), p. 128 - ↑ Igor Minevev, "Submultiplicativity and the Hanna Neumann Conjecture." Ann. of Math., 175 (2012), no. 1, 393-414.
- ↑ "Sheaves on Graphs and a Proof of the Hanna Neumann Conjecture". Math.ubc.ca. Retrieved 2012-06-12.
- ↑ Wilhelm Magnus,
*Uber freie Faktorgruppen und freie Untergruppen Gegebener Gruppen*, Monatshefte für Mathematik, vol. 47(1939), pp. 307–313. - ↑ Roger C. Lyndon and Paul E. Schupp. Combinatorial Group Theory. Springer-Verlag, New York, 2001. "Classics in Mathematics" series, reprint of the 1977 edition. ISBN 978-3-540-41158-1; Proposition 5.11, p. 107
- ↑ W. W. Boone.
*Decision problems about algebraic and logical systems as a whole and recursively enumerable degrees of unsolvability.*1968 Contributions to Math. Logic (Colloquium, Hannover, 1966) pp. 13 33 North-Holland, Amsterdam - ↑ Charles F. Miller, III.
*Decision problems for groups — survey and reflections.*Algorithms and classification in combinatorial group theory (Berkeley, CA, 1989), pp. 1–59, Math. Sci. Res. Inst. Publ., 23, Springer, New York, 1992; ISBN 0-387-97685-X - ↑ John Lennox, and Derek J. S. Robinson.
*The theory of infinite soluble groups.*Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2004. ISBN 0-19-850728-3 - ↑ G. Baumslag, C. F. Miller and H. Short.
*Unsolvable problems about small cancellation and word hyperbolic groups.*Bulletin of the London Mathematical Society, vol. 26 (1994), pp. 97–101 - ↑ Ilya Kapovich, and Richard Weidmann.
*Kleinian groups and the rank problem*. Geometry and Topology, vol. 9 (2005), pp. 375–402 - ↑ John R. Stallings.
*Problems about free quotients of groups.*Geometric group theory (Columbus, OH, 1992), pp. 165–182, Ohio State Univ. Math. Res. Inst. Publ., 3, de Gruyter, Berlin, 1995. ISBN 3-11-014743-2 - ↑ A. A. Razborov.
*Systems of equations in a free group.*(in Russian) Izvestia Akademii Nauk SSSR, Seriya Matematischeskaya, vol. 48 (1984), no. 4, pp. 779–832. - ↑ G. S.Makanin
*Equations in a free group.*(Russian), Izvestia Akademii Nauk SSSR, Seriya Matematischeskaya, vol. 46 (1982), no. 6, pp. 1199–1273 - ↑ Shelly L. Harvey.
*On the cut number of a 3-manifold.*Geometry & Topology, vol. 6 (2002), pp. 409–424 - ↑ Aschbacher, M. (2002),
*Finite Group Theory*, Cambridge University Press, p. 5, ISBN 978-0-521-78675-1

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In abstract algebra, an **abelian group**, also called a **commutative group**, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, these are the groups that obey the axiom of commutativity. Abelian groups generalize the arithmetic of addition of integers. They are named after early 19th century mathematician Niels Henrik Abel.

A **quotient group** or **factor group** is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves the group structure. For example, the cyclic group of addition modulo *n* can be obtained from the integers by identifying elements that differ by a multiple of *n* and defining a group structure that operates on each such class as a single entity. It is part of the mathematical field known as group theory.

In mathematics, one method of defining a group is by a **presentation**. One specifies a set *S* of **generators** so that every element of the group can be written as a product of powers of some of these generators, and a set *R* of **relations** among those generators. We then say *G* has presentation

In mathematics, an **algebraic torus** is a type of commutative affine algebraic group. These groups were named by analogy with the theory of *tori* in Lie group theory.

In geometric group theory, **Gromov's theorem on groups of polynomial growth**, first proved by Mikhail Gromov, characterizes finitely generated groups of *polynomial* growth, as those groups which have nilpotent subgroups of finite index.

In geometric topology, a field within mathematics, the obstruction to a homotopy equivalence
of finite CW-complexes being a simple homotopy equivalence is its **Whitehead torsion**
which is an element in the **Whitehead group**
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In mathematics, a group is **supersolvable** if it has an invariant normal series where all the factors are cyclic groups. Supersolvability is stronger than the notion of solvability.

In mathematics, in the realm of group theory, a countable group is said to be **SQ-universal** if every countable group can be embedded in one of its quotient groups. SQ-universality can be thought of as a measure of largeness or complexity of a group.

In Lie theory and related areas of mathematics, a **lattice** in a locally compact group is a discrete subgroup with the property that the quotient space has finite invariant measure. In the special case of subgroups of **R**^{n}, this amounts to the usual geometric notion of a lattice as a periodic subset of points, and both the algebraic structure of lattices and the geometry of the space of all lattices are relatively well understood.

**Bass–Serre theory** is a part of the mathematical subject of group theory that deals with analyzing the algebraic structure of groups acting by automorphisms on simplicial trees. The theory relates group actions on trees with decomposing groups as iterated applications of the operations of free product with amalgamation and HNN extension, via the notion of the **fundamental group of a graph of groups.** Bass–Serre theory can be regarded as one-dimensional version of the orbifold theory.

**John Robert Stallings Jr.** was a mathematician known for his seminal contributions to geometric group theory and 3-manifold topology. Stallings was a Professor Emeritus in the Department of Mathematics at the University of California at Berkeley where he had been a faculty member since 1967. He published over 50 papers, predominantly in the areas of geometric group theory and the topology of 3-manifolds. Stallings' most important contributions include a proof, in a 1960 paper, of the Poincaré Conjecture in dimensions greater than six and a proof, in a 1971 paper, of the Stallings theorem about ends of groups.

In the mathematical subject of group theory, the **Hanna Neumann conjecture** is a statement about the rank of the intersection of two finitely generated subgroups of a free group. The conjecture was posed by Hanna Neumann in 1957.
In 2011, a strengthened version of the conjecture was proved independently by Joel Friedman
and by Igor Mineyev.

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