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The group Sz(2) is solvable and is the Frobenius group of order 20. The Suzuki groups Sz(q) have orders q 2 (q 2 +1)(q−1). These groups have orders divisible by 5, but not by 3. The Schur multiplier is trivial for n>1, Klein 4-group for n=1, i. e. Sz(8). The outer automorphism group is cyclic of order 2n+1, given by automorphisms of the field ...
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The group (Z,+) of integers is free of rank 1; a generating set is S = {1}.The integers are also a free abelian group, although all free groups of rank are non-abelian. A free group on a two-element set S occurs in the proof of the Banach–Tarski paradox and is described there.
The 5th roots of unity in the complex plane form a group under multiplication. Each non-identity element generates the group. In abstract algebra, a generating set of a group is a subset of the group set such that every element of the group can be expressed as a combination (under the group operation) of finitely many elements of the subset and their inverses.
In less formal terms, the group consists of words in the generators and their inverses, subject only to canceling a generator with an adjacent occurrence of its inverse. If G is any group, and S is a generating subset of G, then every element of G is also of the above form; but in general, these products will not uniquely describe an element of G.
The irreducible complex characters of a finite group form a character table which encodes much useful information about the group G in a concise form. Each row is labelled by an irreducible character and the entries in the row are the values of that character on any representative of the respective conjugacy class of G (because characters are class functions).
The symmetry group of a cube is the internal direct product of the subgroup of rotations and the two-element group {−I, I}, where I is the identity element and −I is the point reflection through the center of the cube. A similar fact holds true for the symmetry group of an icosahedron. Let n be odd, and let D 4n be the dihedral group of ...
In mathematics, the classification of finite simple groups (popularly called the enormous theorem [1] [2]) 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 group ...