Group (math) | Wikipedia audio article

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00:00:20 1 History
00:00:41 2 Definition and illustration
00:01:02 2.1 First example: the integers
00:01:12 2.2 Definition
00:01:33 2.3 Second example: a symmetry group
00:01:54 3 Elementary consequences of the group axioms
00:02:14 3.1 Uniqueness of identity element and inverses
00:02:35 3.2 Division
00:02:56 4 Basic concepts
00:03:16 4.1 Group homomorphisms
00:03:37 4.2 Subgroups
00:03:58 4.3 Cosets
00:04:19 4.4 Quotient groups
00:04:39 5 Examples and applications
00:05:00 5.1 Numbers
00:05:21 5.1.1 Integers
00:05:42 5.1.2 Rationals
00:06:02 5.2 Modular arithmetic
00:06:13 5.3 Cyclic groups
00:06:33 5.4 Symmetry groups
00:06:54 5.5 General linear group and representation theory
00:07:15 5.6 Galois groups
00:07:36 6 Finite groups
00:07:56 6.1 Classification of finite simple groups
00:08:17 7 Groups with additional structure
00:08:38 7.1 Topological groups
00:08:59 7.2 Lie groups
00:09:19 8 Generalizations
00:09:40 9 See also
00:10:01 10 Notes
00:10:22 11 Citations
00:10:42 12 References
00:11:03 12.1 General references
00:11:24 12.2 Special references
00:11:34 12.3 Historical references
00:11:55 13 External links
00:12:16 {gN, g ∈ G}, "G modulo N".This set inherits a group operation (sometimes called coset multiplication, or coset addition) from the original group G: (gN) • (hN)
00:12:36 N serves as the identity in this group, and the inverse of gN in the quotient group is (gN)−1
00:12:57 fvR. The group operation on the quotient is shown at the right. For example, U • U
00:13:18 (fv • fv)R
00:13:39 r1, the right rotation and f
00:13:59 f 2
00:14:20 Examples and applications
00:14:41 Numbers
00:15:02 Integers
00:15:22 2 is an integer, but the only solution to the equation a · b
00:15:43 Rationals
00:16:14 Modular arithmetic
00:16:35 5, there are four group elements 1, 2, 3, 4. In this group, 4 · 4
00:16:56 5 above, the inverse of 4 is 4, and the inverse of 3 is 2, as 3 · 2
00:17:16 Cyclic groups
00:17:37 e, a, a2, a3, ...,where a2 means a • a, and a−3 stands for a−1 • a−1 • a−1
00:17:58 1. These numbers can be visualized as the vertices on a regular n-gon, as shown in blue at the right for n
00:18:19 5, 3 is a generator since 31
00:18:39 Symmetry groups
00:19:00 General linear group and representation theory
00:19:21 Galois groups
00:19:41 Finite groups
00:20:02 Classification of finite simple groups
00:20:23 23 above shows. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups. An intermediate step is the classification of finite simple groups. A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself. The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group—the "monster group"—and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena.
00:20:44 Groups with additional structure
00:20:54 Many groups are simultaneously groups and examples of other mathematical structures. In the language of category theory, they are group objects in a category, meaning that they are objects (that is, examples of another mathematical structure) which come with transformations (called morphisms) that mimic the group axioms. For example, every group (as defined above) is also a set, so a group is a group object in the category of sets.
00:21:15 Topological groups
00:21:36 Lie groups
00:21:56 Generalizations
00:22:17 See also
00:22:27 Notes



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