mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...

, the Hall–Higman theorem, due to , describes the possibilities for the minimal polynomial of an element of

prime power In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number. For example: , and are prime powers, while , and are not. The sequence of prime powers begins: 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 1 ...

order for a

representation Representation may refer to: Law and politics *Representation (politics), political activities undertaken by elected representatives, as well as other theories ** Representative democracy, type of democracy in which elected officials represent a ...

of a ''p''-solvable group.

Statement

Suppose that ''G'' is a ''p''-solvable group with no normal ''p''-subgroups, acting faithfully on a

vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...

over a field of characteristic ''p''. If ''x'' is an element of order ''p''^''n'' of ''G'' then the minimal polynomial is of the form (''X'' − 1)^''r'' for some ''r'' ≤ ''p''^''n''. The Hall–Higman theorem states that one of the following 3 possibilities holds: *''r'' = ''p''^''n'' *''p'' is a

Fermat prime In mathematics, a Fermat number, named after Pierre de Fermat (1601–1665), the first known to have studied them, is a positive integer of the form:F_ = 2^ + 1, where ''n'' is a non-negative integer. The first few Fermat numbers are: 3, 5, ...

and the

Sylow 2-subgroup In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fixed ...

s of ''G'' are non-abelian and ''r'' ≥ ''p''^''n'' −''p''^''n''−1 *''p'' = 2 and the Sylow ''q''-subgroups of ''G'' are non-abelian for some

Mersenne prime In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 1 ...

''q'' = 2^''m'' − 1 less than 2^''n'' and ''r'' ≥ 2^''n'' − 2^{''n''−''m''}.

Examples

The group SL₂(F₃) is 3-solvable (in fact solvable) and has an obvious 2-dimensional representation over a field of characteristic ''p''=3, in which the elements of order 3 have minimal polynomial (''X''−1)² with ''r''=3−1.

References

* * {{DEFAULTSORT:Hall-Higman theorem Theorems in group theory Number theory