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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 objects. They were the first sporadic groups to be discovered. Sometimes the notation ''M''9, ''M''10, ''M''20 and ''M''21 is used for related groups (which act on sets of 9, 10, 20, and 21 points, respectively), namely the stabilizers of points in the larger groups. While these are not sporadic simple groups, they are subgroups of the larger groups and can be used to construct the larger ones. John Conway has shown that one can also extend this sequence up, obtaining the Mathieu groupoid ''M''13 acting on 13 points. ''M''21 is simple, but is not a sporadic group, being isomorphic to PSL(3,4). History introduced the group ''M''12 as part of an investigation of multiply transitive permutation groups, and briefly mentioned (on page 27 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathieu Group M23
In the area of modern algebra known as group theory, the Mathieu group ''M''23 is a sporadic simple group of order : 2732571123 = 10200960 : ≈ 1 × 107. History and properties ''M''23 is one of the 26 sporadic groups and was introduced by . It is a 4-fold transitive permutation group on 23 objects. The Schur multiplier and the outer automorphism group are both trivial. calculated the integral cohomology, and showed in particular that M23 has the unusual property that the first 4 integral homology groups all vanish. The inverse Galois problem seems to be unsolved for M23. In other words, no polynomial in Z 'x''seems to be known to have M23 as its Galois group. The inverse Galois problem is solved for all other sporadic simple groups. Construction using finite fields Let be the finite field with 211 elements. Its group of units has order − 1 = 2047 = 23 · 89, so it has a cyclic subgroup of order 23. The Mathieu group M23 can be identified ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathieu Group M24
In the area of modern algebra known as group theory, the Mathieu group ''M24'' is a sporadic simple group of order : 21033571123 = 244823040 : ≈ 2. History and properties ''M24'' is one of the 26 sporadic groups and was introduced by . It is a 5-transitive permutation group on 24 objects. The Schur multiplier and the outer automorphism group are both trivial. The Mathieu groups can be constructed in various ways. Initially, Mathieu and others constructed them as permutation groups. It was difficult to see that M24 actually existed, that its generators did not just generate the alternating group A24. The matter was clarified when Ernst Witt constructed M24 as the automorphism (symmetry) group of an S(5,8,24) Steiner system W24 (the Witt design). M24 is the group of permutations that map every block in this design to some other block. The subgroups M23 and M22 then are easily defined to be the stabilizers of a single point and a pair of points respe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Theory
In 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. Various physical systems, such as crystals and the hydrogen atom, and three of the four known fundamental forces in the universe, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also central to public key cryptography. The early history of group theory dates from the 19th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jordan's Theorem (symmetric Group)
In finite group theory, Jordan's theorem states that if a primitive permutation group ''G'' is a subgroup of the symmetric group ''S''''n'' and contains a ''p''- cycle for some prime number ''p'' < ''n'' − 2, then ''G'' is either the whole symmetric group ''S''''n'' or the ''A''''n''. It was first proved by Camille Jordan. The statement can be generalized to the case that ''p'' is a 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 sequenc ...
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Automorphisms Of The Symmetric And Alternating Groups
In group theory, a branch of mathematics, the automorphisms and outer automorphisms of the symmetric groups and alternating groups are both standard examples of these automorphisms, and objects of study in their own right, particularly the exceptional outer automorphism of S6, the symmetric group on 6 elements. Summary Generic case * n\neq 2,6: \operatorname(\mathrm_n) = \mathrm_n, and thus \operatorname(\mathrm_n) = \mathrm_1. :Formally, \mathrm_n is complete and the natural map \mathrm_n \to \operatorname(\mathrm_n) is an isomorphism. * n\neq 1,2,6: \operatorname(\mathrm_n)=\mathrm_n/\mathrm_n=\mathrm_2, and the outer automorphism is conjugation by an odd permutation. * n\neq 2,3,6: \operatorname(\mathrm_n)=\operatorname(\mathrm_n)=\mathrm_n :Indeed, the natural maps \mathrm_n \to \operatorname(\mathrm_n) \to \operatorname(\mathrm_n) are isomorphisms. Exceptional cases * n=1,2: trivial: :: \operatorname(\mathrm_1)=\operatorname(\mathrm_1)=\operatorname(\mathrm_1)=\operato ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutator Subgroup
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important because it is the smallest normal subgroup such that the quotient group of the original group by this subgroup is abelian. In other words, G/N is abelian if and only if N contains the commutator subgroup of G. So in some sense it provides a measure of how far the group is from being abelian; the larger the commutator subgroup is, the "less abelian" the group is. Commutators For elements g and h of a group ''G'', the commutator of g and h is ,h= g^h^gh. The commutator ,h/math> is equal to the identity element ''e'' if and only if gh = hg , that is, if and only if g and h commute. In general, gh = hg ,h/math>. However, the notation is somewhat arbitrary and there is a non-equivalent variant definition for the commutator that has the inverses on the right hand side of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod when is a prime number. The ''order'' of a finite field is its number of elements, which is either a prime number or a prime power. For every prime number and every positive integer there are fields of order p^k, all of which are isomorphic. Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory. Properties A finite field is a finite set which is a field; this means that multiplication, addition, subtraction and division (excluding division by zero ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quaternion Group
In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset \ of the quaternions under multiplication. It is given by the group presentation :\mathrm_8 = \langle \bar,i,j,k \mid \bar^2 = e, \;i^2 = j^2 = k^2 = ijk = \bar \rangle , where ''e'' is the identity element and commutes with the other elements of the group. Another presentation of Q8 is :\mathrm_8 = \langle a,b \mid a^4 = e, a^2 = b^2, ba = a^b\rangle. Compared to dihedral group The quaternion group Q8 has the same order as the dihedral group D4, but a different structure, as shown by their Cayley and cycle graphs: In the diagrams for D4, the group elements are marked with their action on a letter F in the defining representation R2. The same cannot be done for Q8, since it has no faithful representation in R2 or R3. D4 can be realized as a subset of the split-quaternions in the same way that Q8 can be viewed as a s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Special Unitary Group
In mathematics, the projective unitary group is the quotient of the unitary group by the right multiplication of its center, , embedded as scalars. Abstractly, it is the holomorphic isometry group of complex projective space, just as the projective orthogonal group is the isometry group of real projective space. In terms of matrices, elements of are complex unitary matrices, and elements of the center are diagonal matrices equal to multiplied by the identity matrix. Thus, elements of correspond to equivalence classes of unitary matrices under multiplication by a constant phase . Abstractly, given a Hermitian space , the group is the image of the unitary group in the automorphism group of the projective space . Projective special unitary group The projective special unitary group PSU() is equal to the projective unitary group, in contrast to the orthogonal case. The connections between the U(), SU(), their centers, and the projective unitary groups is shown at right. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternating Group
In mathematics, an alternating group is the 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 by or Basic properties For , the group A''n'' is the commutator subgroup of the symmetric group S''n'' with index 2 and has therefore ''n''!/2 elements. It is the kernel of the signature group homomorphism explained under symmetric group. The group A''n'' is abelian if and only if and simple if and only if or . A5 is the smallest non-abelian simple group, having order 60, and the smallest non- solvable group. The group A4 has the Klein four-group V as a proper normal subgroup, namely the identity and the double transpositions , that is the kernel of the surjection of A4 onto . We have the exact sequence . In Galois theory, this map, or rather the corresponding map , corresponds to associating the Lagrange resolvent cubic to a quartic, w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost Simple Group
In mathematics, a group is said to be almost simple if it contains a non-abelian simple group and is contained within the automorphism group of that simple group – that is, if it fits between a (non-abelian) simple group and its automorphism group. In symbols, a group ''A'' is almost simple if there is a (non-abelian) simple group ''S'' such that S \leq A \leq \operatorname(S). Examples * Trivially, non-abelian simple groups and the full group of automorphisms are almost simple, but proper examples exist, meaning almost simple groups that are neither simple nor the full automorphism group. * For n=5 or n \geq 7, the symmetric group \mathrm_n is the automorphism group of the simple alternating group \mathrm_n, so \mathrm_n is almost simple in this trivial sense. * For n=6 there is a proper example, as \mathrm_6 sits properly between the simple \mathrm_6 and \operatorname(\mathrm_6), due to the exceptional outer automorphism of \mathrm_6. Two other groups, the Mathieu group \mathr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 associated projective space P(''V''). Explicitly, the projective linear group is the quotient group :PGL(''V'') = GL(''V'')/Z(''V'') where GL(''V'') is the general linear group of ''V'' and Z(''V'') is the subgroup of all nonzero scalar transformations of ''V''; these are quotiented out because they act trivially on the projective space and they form the kernel of the action, and the notation "Z" reflects that the scalar transformations form the center of the general linear group. The projective special linear group, PSL, is defined analogously, as the induced action of the special linear group on the associated projective space. Explicitly: :PSL(''V'') = SL(''V'')/SZ(''V'') where SL(''V'') is the special linear group over ''V'' and SZ(''V'') i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
