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Non-abelian group. 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 ...
D 2n: the dihedral group of order 2n, the same as Dih n (notation used in section List of small non-abelian groups) S n: the symmetric group of degree n, containing the n! permutations of n elements; A n: the alternating group of degree n, containing the even permutations of n elements, of order 1 for n = 0, 1, and order n!/2 otherwise
A small example of a solvable, non-nilpotent group is the symmetric group S 3. In fact, as the smallest simple non-abelian group is A 5, (the alternating group of degree 5) it follows that every group with order less than 60 is solvable.
In group theory, the quaternion group Q 8 (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. where e is the identity element and e commutes with the other elements of the group.
Group cohomology plays a role in the investigation of fixed points of a group action in a module or space and the quotient module or space with respect to a group action. Group cohomology is used in the fields of abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory ...
William Burnside (1911, p. 503 note M) conjectured that every nonabelian finite simple group has even order. Richard Brauer () suggested using the centralizers of involutions of simple groups as the basis for the classification of finite simple groups, as the Brauer–Fowler theorem shows that there are only a finite number of finite simple groups with given centralizer of an involution.
e. In mathematics, the classification of finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or belongs to a broad infinite class called the groups of Lie type, or else it is one of twenty-six exceptions, called sporadic. (The Tits group is sometimes regarded as a sporadic ...
For a small non-abelian example, consider the quaternion group Q 8, which is a smallest non-abelian p-group. It has center {1, −1} of order 2, and its upper central series is {1}, {1, −1}, Q 8; so it is nilpotent of class 2. The direct product of two nilpotent groups is nilpotent. [5] All finite p-groups are in fact nilpotent .