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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 universal covering group of the circle group T is the additive group of real numbers (R, +) with the covering homomorphism given by the mapping R → T : x ↦ exp(2πix). The kernel of this mapping is isomorphic to Z. For any integer n we have a covering group of the circle by itself T → T that sends z to z n.
To qualify as an abelian group, the set and operation, (,), must satisfy four requirements known as the abelian group axioms (some authors include in the axioms some properties that belong to the definition of an operation: namely that the operation is defined for any ordered pair of elements of A, that the result is well-defined, and that the ...
Every elementary abelian p-group is a vector space over the prime field with p elements, and conversely every such vector space is an elementary abelian group. By the classification of finitely generated abelian groups, or by the fact that every vector space has a basis, every finite elementary abelian group must be of the form (Z/pZ) n for n a ...
If G is a finite non-abelian simple group (e.g. G = A n, the alternating group, for n > 4) then rank(G) = 2. This fact is a consequence of the Classification of finite simple groups . If G is a finitely generated group and Φ( G ) ≤ G is the Frattini subgroup of G (which is always normal in G so that the quotient group G /Φ( G ) is defined ...
Abelian varieties appear naturally as Jacobian varieties (the connected components of zero in Picard varieties) and Albanese varieties of other algebraic varieties. The group law of an abelian variety is necessarily commutative and the variety is non-singular. An elliptic curve is an abelian variety of dimension 1.
The free abelian group on S can be explicitly identified as the free group F(S) modulo the subgroup generated by its commutators, [F(S), F(S)], i.e. its abelianisation. In other words, the free abelian group on S is the set of words that are distinguished only up to the order of letters. The rank of a free group can therefore also be defined as ...
Members of the same conjugacy class cannot be distinguished by using only the group structure, and therefore share many properties. The study of conjugacy classes of non-abelian groups is fundamental for the study of their structure. [1] [2] For an abelian group, each conjugacy class is a set containing one element (singleton set).