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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
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 ...
Representation theory of SU (2) In the study of the representation theory of Lie groups, the study of representations of SU (2) is fundamental to the study of representations of semisimple Lie groups. It is the first case of a Lie group that is both a compact group and a non-abelian group. The first condition implies the representation theory ...
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.
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.
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.
It is the kernel of the signature group homomorphism sgn : S n → {1, −1} explained under symmetric group. The group A n is abelian if and only if n ≤ 3 and simple if and only if n = 3 or n ≥ 5. A 5 is the smallest non-abelian simple group, having order 60, and the smallest non-solvable group.