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In mathematics, and specifically in group theory, a non-abelian group, sometimes called a non-commutative group, is a group (G, ∗) in which there exists at least one pair of elements a and b of G, such that a ∗ b ≠ b ∗ a. [1] [2] This class of groups contrasts with the abelian groups, where all pairs of group elements commute.
The representation theory of the symmetric group is a particular case of the representation theory of finite groups, for which a concrete and detailed theory can be obtained. This has a large area of potential applications, from symmetric function theory to problems of quantum mechanics for a number of identical particles .
The smallest nonabelian simple group is the alternating group of order 60, and every simple group of order 60 is isomorphic to . [2] The second smallest nonabelian simple group is the projective special linear group PSL(2,7) of order 168, and every simple group of order 168 is isomorphic to PSL(2,7). [3] [4]
The quaternion group has the unusual property of being Hamiltonian: Q 8 is non-abelian, but every subgroup is normal. [4] Every Hamiltonian group contains a copy of Q 8. [5] The quaternion group Q 8 and the dihedral group D 4 are the two smallest examples of a nilpotent non-abelian group.
The structure of any finite abelian group is relatively simple; every finite abelian group is the direct sum of cyclic p-groups. This can be extended to a complete classification of all finitely generated abelian groups, that is all abelian groups that are generated by a finite set. The situation is much more complicated for the non-abelian ...
Group theory has three main historical sources: number theory, the theory of algebraic equations, and geometry.The number-theoretic strand was begun by Leonhard Euler, and developed by Gauss's work on modular arithmetic and additive and multiplicative groups related to quadratic fields.
A 5 is the smallest non-abelian simple group, having order 60, and thus the smallest non-solvable group. The group A 4 has the Klein four-group V as a proper normal subgroup, namely the identity and the double transpositions { (), (12)(34), (13)(24), (14)(23) }, that is the kernel of the surjection of A 4 onto A 3 ≅ Z 3.
Only the neutral elements are symmetric to the main diagonal, so this group is not abelian. Cayley table as general (and special) linear group GL(2, 2) In mathematics, D 3 (sometimes alternatively denoted by D 6) is the dihedral group of degree 3 and order 6. It equals the symmetric group S 3. It is also the smallest non-abelian group. [1]