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List of all nonabelian groups up to order 31 Order Id. [a] G o i Group Non-trivial proper subgroups [1] Cycle graph Properties 6 7 G 6 1: D 6 = S 3 = Z 3 ⋊ Z 2: Z 3, Z 2 (3) : Dihedral group, Dih 3, the smallest non-abelian group, symmetric group, smallest Frobenius group.
Outer automorphism group Trivial Order 2 Order 2 Trivial Other names J(1), J(11) Hall–Janko group, HJ Higman–Janko–McKay group, HJM Remarks It is a subgroup of G 2 (11), and so has a 7-dimensional representation over the field with 11 elements. The automorphism group J 2:2 of J 2 is the automorphism group of a rank 3 graph on 100 points ...
The rank of a symmetry group is closely related to the complexity of the object (a molecule, a crystal structure) being under the action of the group. If G is a crystallographic point group, then rank(G) is up to 3. [9] If G is a wallpaper group, then rank(G) = 2 to 4. The only wallpaper-group type of rank 4 is p2mm. [10]
A group is called torsion-free if it has no non-trivial torsion elements. The factor-group A/T(A) is the unique maximal torsion-free quotient of A and its rank coincides with the rank of A. The notion of rank with analogous properties can be defined for modules over any integral domain, the case of abelian groups corresponding to modules over Z.
p-groups of the same order are not necessarily isomorphic; for example, the cyclic group C 4 and the Klein four-group V 4 are both 2-groups of order 4, but they are not isomorphic. Nor need a p-group be abelian; the dihedral group Dih 4 of order 8 is a non-abelian 2-group. However, every group of order p 2 is abelian.
The semidirect product of a cyclic group of order p 2 by a cyclic group of order p acting non-trivially on it. This group has exponent p 2. If n is a positive integer there are two extraspecial groups of order p 1+2n, which for p odd are given by The central product of n extraspecial groups of order p 3, all of exponent p.
This is in fact the significance in group-theoretical terms of the unique non-trivial element of (/,),. An example of a second cohomology group is the Brauer group: it is the cohomology of the absolute Galois group of a field k which acts on the invertible elements in a separable closure:
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 ...