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# Topic: Finite group

 Finite group - Wikipedia, the free encyclopedia Some aspects of the theory of finite groups were investigated in great depth in the twentieth century, in particular the local theory, and the theory of solvable groups and nilpotent groups. Finite groups are directly relevant to symmetry, when that is restricted to a finite number of transformations. In this way, finite groups and their properties can enter centrally in questions, for example in theoretical physics, where their role is not initially obvious. en.wikipedia.org /wiki/Finite_group   (243 words)

 Group (mathematics) - Wikipedia, the free encyclopedia Previous to this work, groups were mainly studied concretely, in the form of permutations; some aspects of abelian group theory were known in the theory of quadratic forms. A group that we are introduced to in elementary school is the integers under addition. Given a group G and a normal subgroup N, the quotient group is the set of cosets of G/N together with the operation (gN)(hN)=ghN. en.wikipedia.org /wiki/Group_(mathematics)   (1904 words)

 PlanetMath: locally finite group A locally finite group is a torsion group. Burnside, however, did show that if a matrix group is torsion, then it is locally finite. This is version 3 of locally finite group, born on 2004-04-16, modified 2004-12-10. planetmath.org /encyclopedia/LocallyFiniteGroup.html   (190 words)

 Automatic Groups When the group is a finitely generated group given in terms of generators and relations, the question is known as the word problem. For example, assuming that all elements of a finite group are generators, for any x in the group, the identity word and the word (x)(x^(-1)) represent the same group element. In order to define word-hyperbolic groups, we need to discuss Cayley graphs; the Cayley graph is a graph with a point for each element of the group and with a directed line from one point to a second point whenever the second element is the first element times one of the generators. www.geom.uiuc.edu /docs/forum/automaticgroups   (1192 words)

 20: Group Theory and Generalizations Groups acting on vector spaces are subgroups of the matrix groups studied in Linear Algebra. Groups acting on topological spaces are the basis of equivariant topology and homotopy theory in Algebraic Topology. Nielsen's theorem: subgroups of free groups are free. www.math.niu.edu /~rusin/known-math/index/20-XX.html   (2774 words)

 MTH-3D15 : Theory of Finite Groups Group theory is a very large field which interconnects with many branches of pure and applied mathematics. Abstract groups began to emerge with Jordan's seminal Traité des substitutions et des equations algébriques (1870) while the definition of abstract groups in general appears to be due to Weber (1882). For finite groups the orbit stabilizer theorem, a relatively easy result on group actions, plays a central role and a great many results are a consequence of it. www.mth.uea.ac.uk /maths/syllabuses/0506/3D1505.html   (767 words)

 Group Theory   (Site not responding. Last check: 2007-10-21) When a group is in fact commutative - i.e., for all a and b in the group one has a#b = b#a - then it is said to be an Abelian group. The number of elements of a finite group (G,#) is denoted by the symbol G, and is said to be the The order of this group is 8 - that is, G Consider the set H = {2, 2#2, 2#(2#2), 2#(2#(2#2)),......}, a subset of G. Doing the indicated computations show that H = {2, 4, 8, 1}, and one can check that indeed (H,#) is a subgroup of (G,#). math.boisestate.edu /~marion/teaching/crypto2/group_theory.htm   (970 words)

 ABSTRACT ALGEBRA ON LINE: Groups A group G is said to be a finite group if the set G has a finite number of elements. Let G be a group, and let H be a subset of G. Then H is called a subgroup of G if H is itself a group, under the operation induced by G. Proposition. Any subgroup of the symmetric group Sym(S) on a set S is called a permutation group or group of permutations. www.math.niu.edu /~beachy/aaol/groups.html   (1115 words)

 PlanetMath: character of a finite group "character of a finite group" is owned by alozano. the sum of the values of a character of a finite group is This is version 3 of character of a finite group, born on 2004-02-20, modified 2005-04-25. planetmath.org /encyclopedia/CharacterOfAFiniteGroup.html   (68 words)

 3. Knot and Manifold Groups - Residually Finite Groups Thus the fundamental group of a fibered knot is an extension of a finitely generated free group (the fundamental group of a punctured surface) by an infinite cyclic group. In a later argument, using Schubert's normal form for the group of a two-bridge knot, he constructed a one-relator presentation for the group and showed that it was residually finite [Mayl74]. The fundamental group of a Seifert fibered space is residually finite [Hemp76, pp 177] and those components which have hyperbolic structures have fundamental groups which are subgroups of the modular group of matrices, and hence are residually finite [Mal40]. www.math.umbc.edu /~campbell/CombGpThy/RF_Thesis/3_Knot_Manifold_Groups.html   (1205 words)

 Simple Groups of Finite Morley rank The connected component of a Sylow 2-subgroup of a group of finite Morley rank is a central product U*T with U 2-unipotent and T a 2-torus; the intersection of U and T is finite. More generally, there are no simple groups of finite Morley rank of mixed type, and any simple group of finite Morley rank is algebraic. In the odd type case, any simple K*-group of finite Morley rank is either algebraic, or is minimal connected simple with Prüfer 2-rank at most two. www.rci.rutgers.edu /~cherlin/FMR   (555 words)

 Finite group torsors for the qfh topology, by J.F. Jardine   (Site not responding. Last check: 2007-10-21) Finite group torsors for the qfh topology, by J.F. Jardine This is the non-abelian analogue of a comparison theorem of Voevodsky which identifies etale and qfh cohomology for coefficient sheaves represented by finite etale abelian group schemes. The method of proof is to show that the qfh stack completion of the group scheme G induces a local weak equivalence of classifying simplicial sheaves for the etale topology. www.math.uiuc.edu /K-theory/0461   (77 words)

 Pro-finite group - Wikipedia A pro-finite group is a group which is the inverse limit of finite groups. If we regard each of the finite groups as having the discrete topology, then as a subset of their product, the pro-finite group inherits a topology. Since all of the conditions on an inverse limit are closed in any Hausdorff space, and since the product of compact spaces is compact, any pro-finite group is a compact Hausdorff space, and the group operations are continuous with respect to this topology. nostalgia.wikipedia.org /wiki/Pro-finite_group   (126 words)

 Pro-finite group - Wikipedia   (Site not responding. Last check: 2007-10-21) In mathematics, a pro-finite group G is a group that is the inverse limit of finite groups. Each of the finite groups is regarded as carrying the discrete topology, and since G is a subset of the product of these discrete spaces, it inherits a topology which turns it into a topological group. Since all finite discrete spaces are compact Hausdorff spaces, their product will be a compact Hausdorff space by Tychonoff's theorem. www.mvlife.com /mv/mvlife_wiki/pr/Profinite_group.html   (96 words)

 Group Theory & Rubik's Cube Group theory is the study of the algebra of transformations and symmetry. Given an element x of a group G, the orbit of x is the set of all elements of G which are generated by x, i.e. A representation of a group G is a set of matrices M which are homomorphic to the group. akbar.marlboro.edu /~mahoney/courses/Spr00/rubik.html   (3602 words)

 [No title] Therefore in [D-W] a p-discrete torus of rank n is defined to be a discr* *ete group isomorphic to (Z=p1)n and a p-discrete toral group to be an extension of* * a p-discrete torus by a finite p-group. The motivation of this definition comes from the fact th* *at, for a compact connected Lie group G the maximal torus is self centralizing, and that therefore the centralizer of the maximal torus of a nonconnected compact L* *ie group is always a p-toral group. Hence ss2(BA) is a free Z^p-module and A a p-compact toral group. hopf.math.purdue.edu /Moller-Notbohm/centers.txt   (12228 words)

 20D: Abstract Finite Groups Moreover, we study the internal properties of those groups -- material on their representation (20C) or permutation actions (20B) or cohomology (20J) are for now on the main group theory page. All finite simple groups can be generated by two elements; indeed for alternating (and symmetric) groups, such pairs of generators are legion. Finding all finite groups in which same order elements are in same conjugacy class. www.math.niu.edu /~rusin/known-math/index/20DXX.html   (707 words)

 Math Forum - Ask Dr. Math   (Site not responding. Last check: 2007-10-21) Question: Let G be a finite group, f an automorphism of G such that f^2 is the identity automorphism of G. Suppose that f(x)=x implies that x=e (the identity). Prove that G is abelian and f(a)=a^-1 for all a in G. My professor offered the hint that we should define h(x)=x^-1(f(x)) and show that it is 1-1 and onto, but even that has thrown me through a loop. Since G is finite, if we know that h(x) is injective, it must be surjective as well (this is a property of finite sets) and then the bijective nature of h(x) will have been established. mathforum.org /library/drmath/view/51686.html   (586 words)

 Maybe this Explains the Economic Cycle... best Finite Group   (Site not responding. Last check: 2007-10-21) Examples of finite groups are the modulo multiplication groups, point groups, cyclic groups, dihedral groups, symmetric groups,... Formally, a pro-finite group is a group that is isomorphic to the inverse limit of an inverse system of... Finite Group Theory develops the foundations of the theory of finite groups... ascot.pl /th/Fourier3/Finite-Group.htm   (475 words)

 Class Equation of Finite Group   (Site not responding. Last check: 2007-10-21) PlanetMath: proof that a nontrivial normal subgroup of a finite \$p\$-group \$G\$ an... The distribution of class groups of function fields... Abstracts (Representations of Finite Groups and Related Algebras)... www.scienceoxygen.com /math/269.html   (168 words)

 cannon.html In joint work over the past ten years with Derek Holt he has developed new algorithms for the fundamental invariants of a finite group that makes possible the analysis of groups of a complexity far beyond what was possible previously. The design of efficient algorithms based on the idea of "reduction to a simple group" represents a considerable challenge to the algorithm designer and it is only very recently that efficient practical algorithms have started to emerge. Group Theory Algorithms: G. Butler, "Fundamental Algorithms for Permutation Groups", LNCS 559, Springer, 1991. www.win.tue.nl /math/eidma/courses/minicourses/cannon/cannon.html   (673 words)

 Amazon.com: Books: Finite Group Theory (Cambridge Studies in Advanced Mathematics)   (Site not responding. Last check: 2007-10-21) Many questions about arbitrary groups can be reduced to similar questions about simple groups and applications of the theory are beginning to appear in other branches of mathematics. Finite Group Theory provides the basic background necessary to understand the research literature and apply the theory. Develops the foundations of the theory of finite groups. www.amazon.com /exec/obidos/tg/detail/-/0521458269?v=glance   (595 words)

 Finite Group Exercises in MAPLE It is available from maple stuff (click on the groups link then find the group2.2 link and download it). The Rubik's cube group, corresponding to the set of all possible "states" of the Rubik's cube, is given in MAPLE by The degree and generators and be recovered using the degree_permgroup and gens_permgroup commands. www.ew.usna.edu /~wdj/teach/group22_exercises_v7.html   (907 words)

 groups Examples of Finite Groups: A = {1, -1, i, -i} where * is multiplication, B = {0, 1, 2, 3) where * is addition modulo 4. A subgroup is a group entirely inside another: {1, -1} is a subgroup of A, {0, 2) is is a subgroup of B. The makers of GAP have written an analysis of Rubik's Cube from a Group Theory perspective. www.mathpuzzle.com /groups.html   (581 words)

 The Coset Poset And Probabilistic Zeta Function Of A Finite Group - Brown (ResearchIndex)   (Site not responding. Last check: 2007-10-21) We investigate the topological properties of the poset of proper cosets xH in a finite group G. Of particular interest is the reduced Euler characteristic, which is closely related to the value at \Gamma1 of the probabilistic zeta function of G. Our main result gives divisibility properties of this reduced Euler characteristic. Introduction For a finite group G and a non-negative integer s, let P (G; s) be the probability that a randomly chosen ordered s-tuple from G generates G. Philip... K.S. Brown, The coset poset and probabilistic zeta function of a finite group. citeseer.lcs.mit.edu /brown99coset.html   (465 words)

 Diamond Theory: Symmetry in Binary Spaces   (Site not responding. Last check: 2007-10-21) For example, the affine group A on the 4-space over the 2-element field has a natural noncontinuous and asymmetric but symmetry-preserving action on the elements of a 4x4 array. In the 4x4 case, D is a four-diamond figure (left, below) and G is a group of 322,560 permutations generated by arbitrarily mixing random permutations of rows and of columns with random permutations of the four 2x2 quadrants. A table of the octahedral group O using the 24 patterns from the 2x2 case of the diamond theorem. m759.freeservers.com   (1917 words)

 5. Open Questions - Residually Finite Groups   (Site not responding. Last check: 2007-10-21) Is a finitely generated subgroup of a free soluble group finitely separable? Suppose G and H are finitely generated residually finite groups with the same set of finite homomorphic images. Suppose that G is a residually finite group with one defining relation such that each subgroup of finite index also has one defining relation. www.math.umbc.edu /~campbell/CombGpThy/RF_Thesis/5_Open_Questions.html   (390 words)

 Generic Idempotent Modules For A Finite Group - Benson, Krause (ResearchIndex)   (Site not responding. Last check: 2007-10-21) Let G be a finite group and k an algebraically closed field of characteristic p. We show that FU is a finite sum of generic modules corresponding to the irreducible components of VG. Benson and H. Krause, Generic idempotent modules for a finite group, Preprint, Universitat Bielefeld, 1999. citeseer.ist.psu.edu /benson99generic.html   (428 words)

 Homotopy finite group theory   (Site not responding. Last check: 2007-10-21) A Frobenius system over a finite p-group S is a category whose objects are the subgroups of S and whose morphisms are monomorphisms of groups which include those monomorphisms induced by conjugation in S. A Frobenius system over S is called saturated if it satisfies certain axioms formultated by Lluis Puig, and motivated by the properties of fusion within a finite group G with Sylow p-subgroup S. which is injective on objects and surjective on morphism sets, and which also satisfies certain axioms motivated by the case of finite groups. www.maths.abdn.ac.uk /~stc2001/abstracts/Oliver/Oliver.html   (174 words)

 Permutation Group Problems A base for a permutation group G is a sequence of points whose pointwise stabiliser is the identity; it is irredundant if no point in the sequence is fixed by the stabiliser of its predecessors. The Parker vector of a finite permutation group G is the n-tuple whose kth component is the number of orbits of G on the set of k-cycles occurring in elements of G. a 2-transitive group), is it a Frobenius group (resp. www.maths.qmw.ac.uk /~pjc/pgprob.html   (2622 words)

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