PSL(2, 7)
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mathematics 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 ...
, the
projective special linear group In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associa ...
, isomorphic to , is a
finite Finite may refer to: * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked for person and/or tense or aspect * "Finite", a song by Sara Gr ...
simple group SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service. The d ...
that has important applications in
algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
,
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, and
number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
. It is the
automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
of the
Klein quartic In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus with the highest possible order automorphism group for this genus, namely order orientation-preserving automorphisms, and automorphisms i ...
as well as the
symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...
of the
Fano plane In finite geometry, the Fano plane (named after Gino Fano) is a finite projective plane with the smallest possible number of points and lines: 7 points and 7 lines, with 3 points on every line and 3 lines through every point. These points and ...
. With 168 elements, PSL(2, 7) is the smallest nonabelian
simple group SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service. The d ...
after the
alternating group In mathematics, an alternating group is the Group (mathematics), group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted ...
A5 with 60 elements, isomorphic to .


Definition

The
general linear group In mathematics, the general linear group of degree n is the set of n\times n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again inve ...
consists of all invertible 2×2
matrices Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the ...
over F7, the
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...
with 7 elements. These have nonzero determinant. The
subgroup In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation  ...
consists of all such matrices with unit
determinant In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...
. Then is defined to be the
quotient group 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 some of the group structure (the rest of the structure is "factored out"). For ex ...
: SL(2, 7) / obtained by identifying ''I'' and −''I'', where ''I'' is the
identity matrix In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. It has unique properties, for example when the identity matrix represents a geometric transformation, the obje ...
. In this article, we let ''G'' denote any group that is isomorphic to .


Properties

''G'' = has 168 elements. This can be seen by counting the possible columns; there are possibilities for the first column, then possibilities for the second column. We must divide by to force the determinant equal to one, and then we must divide by 2 when we identify ''I'' and −''I''. The result is . It is a general result that is
simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by John ...
for (''q'' being some power of a prime number), unless or . is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
to the
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...
S3, and is isomorphic to
alternating group In mathematics, an alternating group is the Group (mathematics), group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted ...
A4. In fact, is the second smallest nonabelian simple group, after the
alternating group In mathematics, an alternating group is the Group (mathematics), group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted ...
. The number of
conjugacy class In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other ...
es and
irreducible representation In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W, ...
s is 6. The sizes of conjugacy classes are 1, 21, 42, 56, 24, 24. The dimensions of irreducible representations 1, 3, 3, 6, 7, 8. Character table : \begin & 1A_ & 2A_ & 4A_ & 3A_ & 7A_ & 7B_ \\ \hline \chi_1 & 1 & 1 & 1 & 1 & 1 & 1 \\ \chi_2 & 3 & -1 & 1 & 0 & \sigma & \bar \sigma \\ \chi_3 & 3 & -1 & 1 & 0 & \bar \sigma & \sigma \\ \chi_4 & 6 & 2 & 0 & 0 & -1 & -1 \\ \chi_5 & 7 & -1 &-1 & 1 & 0 & 0 \\ \chi_6 & 8 & 0 & 0 & -1 & 1 & 1 \\ \end, where : \sigma = \frac. The following table describes the conjugacy classes in terms of the order of an element in the class, the size of the class, the minimum polynomial of every representative in GL(3, 2), and the function notation for a representative in PSL(2, 7). Note that the classes 7A and 7B are exchanged by an automorphism, so the representatives from GL(3, 2) and PSL(2, 7) can be switched arbitrarily. The order of group is , this implies existence of Sylow's subgroups of orders 3, 7 and 8. It is easy to describe the first two, they are cyclic, since any group of prime order is cyclic. Any element of conjugacy class 3''A''56 generates Sylow 3-subgroup. Any element from the conjugacy classes 7''A''24, 7''B''24 generates the Sylow 7-subgroup. The Sylow 2-subgroup is a
dihedral group of order 8 In mathematics, D4 (sometimes alternatively denoted by D8) is the dihedral group of degree 4 and order 8. It is the symmetry group of a square. Symmetries of a square As an example, consider a square of a certain thickness with the letter "F" ...
. It can be described as
centralizer In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set \operatorname_G(S) of elements of ''G'' that commute with every element of ''S'', or equivalently, the set of ele ...
of any element from the conjugacy class 2''A''21. In the representation, a Sylow 2-subgroup consists of the upper triangular matrices. This group and its Sylow 2-subgroup provide a counter-example for various
normal p-complement In group theory, a branch of mathematics, a normal ''p''-complement of a finite group, finite group (mathematics), group for a prime number, prime ''p'' is a normal subgroup of order (group theory), order coprime to ''p'' and index of a subgroup, i ...
theorems for .


Actions on projective spaces

''G'' = acts via
linear fractional transformation In mathematics, a linear fractional transformation is, roughly speaking, an inverse function, invertible transformation of the form : z \mapsto \frac . The precise definition depends on the nature of , and . In other words, a linear fractional t ...
on the
projective line In projective geometry and mathematics more generally, a projective line is, roughly speaking, the extension of a usual line by a point called a '' point at infinity''. The statement and the proof of many theorems of geometry are simplified by the ...
P1(7) over the field with 7 elements: :\text \gamma = \begin a & b \\ c & d \end \in \text(2, 7) \text x \in \mathbf^1\!(7),\ \gamma \cdot x = \frac . Every orientation-preserving automorphism of P1(7) arises in this way, and so can be thought of geometrically as a group of symmetries of the projective line P1(7); the full group of possibly orientation-reversing projective linear automorphisms is instead the order 2 extension , and the group of
collineation In projective geometry, a collineation is a one-to-one and onto map (a bijection) from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear. A collineation is ...
s of the projective line is the complete
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...
of the points. However, is also
isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
to (), the special (general) linear group of 3×3 matrices over the field with 2 elements. In a similar fashion, acts on the
projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane (geometry), plane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, paral ...
P2(2) over the field with 2 elements — also known as the Fano plane: : For \gamma = \begin a & b & c \\ d & e & f \\ g & h & i \end \in \text(3, 2)\ \ and \ \ \mathbf = \begin x \\ y \\ z \end \in \mathbf^2\!(2),\ \ \gamma \ \cdot \ \mathbf = \begin ax+by+cz \\ dx+ey+fz \\ gx+hy+iz \end Again, every automorphism of P2(2) arises in this way, and so can be thought of geometrically as the
symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...
of this projective plane. The
Fano plane In finite geometry, the Fano plane (named after Gino Fano) is a finite projective plane with the smallest possible number of points and lines: 7 points and 7 lines, with 3 points on every line and 3 lines through every point. These points and ...
can be used to describe multiplication of
octonions In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions have ...
, so ''G'' acts on the set of octonion multiplication tables.


Symmetries of the Klein quartic

The
Klein quartic In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus with the highest possible order automorphism group for this genus, namely order orientation-preserving automorphisms, and automorphisms i ...
is the projective variety over the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s C defined by the quartic polynomial : ''x''3''y'' + ''y''3''z'' + ''z''3''x'' = 0. It is a compact
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
of genus , and is the only such surface for which the size of the conformal automorphism group attains the maximum of . This bound is due to the Hurwitz automorphisms theorem, which holds for all . Such " Hurwitz surfaces" are rare; the next genus for which any exist is , and the next after that is . As with all Hurwitz surfaces, the Klein quartic can be given a metric of constant negative curvature and then tiled with
regular Regular may refer to: Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instruments, tunings with equal intervals between the paired notes of successive open strings Other uses * Regular character, ...
(hyperbolic)
heptagon In geometry, a heptagon or septagon is a seven-sided polygon or 7-gon. The heptagon is sometimes referred to as the septagon, using ''Wikt:septa-, septa-'' (an elision of ''Wikt:septua-, septua-''), a Latin-derived numerical prefix, rather than ...
s, as a quotient of the order-3 heptagonal tiling, with the symmetries of the surface as a Riemannian surface or algebraic curve exactly the same as the symmetries of the tiling. For the Klein quartic this yields a tiling by 24 heptagons, and the order of ''G'' is thus related to the fact that . Dually, it can be tiled with 56 equilateral triangles, with 24 vertices, each of degree 7, as a quotient of the
order-7 triangular tiling In geometry, the order-7 triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of . Hurwitz surfaces The symmetry group of the tiling is the (2,3,7) triangle group, and a fundamental domain for this action is th ...
. Klein's quartic arises in many fields of mathematics, including representation theory, homology theory, octonion multiplication,
Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive number, positive integers , , and satisfy the equation for any integer value of greater than . The cases ...
, and Stark's theorem on imaginary quadratic number fields of class number 1.


Mathieu group

is a maximal subgroup of the
Mathieu group In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups ''M''11, ''M''12, ''M''22, ''M''23 and ''M''24 introduced by . They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 objec ...
M21; the groups M21 and M24 can be constructed as extensions of . These extensions can be interpreted in terms of the tiling of the Klein quartic, but are not realized by geometric symmetries of the tiling.


Permutation actions

The group acts on various finite sets: * In its original interpretation as , orientation-preserving linear automorphisms of the projective line P1(F7), it acts transitively on the 8 points with a stabilizer of order 21 fixing a given point. It also acts 2-transitively with stabilizer of order 3 on each pair of points; and it has two orbits on triples of points, with trivial stabilizer on each triple. (The larger group acts sharply 3-transitively.) * Interpreted as , linear automorphisms of the Fano plane P2(F2), it acts 2-transitively on the 7 points, with stabilizer of order 24 fixing each point, and stabilizer of order 4 fixing each pair of points. * Interpreted as automorphisms of a tiling of the Klein quartic, it acts transitively on the 24 vertices (or dually, 24 heptagons), with stabilizer of order 7 (corresponding to a rotation about the vertex/heptagon). * Interpreted as a subgroup of the Mathieu group M21, the subgroup acts non-transitively on 21 points.


References

*


Further reading

* {{cite journal , last1=Brown , first1=Ezra , last2=Loehr , first2=Nicholas , title=Why is PSL (2,7)≅ GL (3,2)? , zbl=1229.20046 , journal=Am. Math. Mon. , volume=116 , number=8 , pages=727–732 , year=2009 , doi=10.4169/193009709X460859 , url=http://www.math.vt.edu/people/brown/doc/PSL%282,7%29_GL%283,2%29.pdf , access-date=2014-09-27 , archive-url=https://web.archive.org/web/20161009000204/http://www.math.vt.edu/people/brown/doc/PSL%282,7%29_GL%283,2%29.pdf , archive-date=2016-10-09 , url-status=dead


External links


The Eightfold Way: the Beauty of Klein's Quartic Curve (Silvio Levy, ed.)


* ttp://www.msri.org/publications/books/Book35/files/elkies.pdf The Klein Quartic in Number Theory (Noam Elkies)
Projective special linear group: PSL(3, 2)
Finite groups Projective geometry Octonions