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  2. Rank of a group - Wikipedia

    en.wikipedia.org/wiki/Rank_of_a_group

    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]

  3. Rank of an abelian group - Wikipedia

    en.wikipedia.org/wiki/Rank_of_an_abelian_group

    Abelian groups of rank 0 are exactly the periodic abelian groups. The group Q of rational numbers has rank 1. Torsion-free abelian groups of rank 1 are realized as subgroups of Q and there is a satisfactory classification of them up to isomorphism. By contrast, there is no satisfactory classification of torsion-free abelian groups of rank 2. [2]

  4. Group (periodic table) - Wikipedia

    en.wikipedia.org/wiki/Group_(periodic_table)

    In the periodic table of the elements, each column is a group. In chemistry, a group (also known as a family) [1] is a column of elements in the periodic table of the chemical elements. There are 18 numbered groups in the periodic table; the 14 f-block columns, between groups 2 and 3, are not numbered.

  5. Periodic table - Wikipedia

    en.wikipedia.org/wiki/Periodic_table

    Periodic table of the chemical elements showing the most or more commonly named sets of elements (in periodic tables), and a traditional dividing line between metals and nonmetals. The f-block actually fits between groups 2 and 3; it is usually shown at the foot of the table to save horizontal space.

  6. Nilpotent group - Wikipedia

    en.wikipedia.org/wiki/Nilpotent_group

    If a group has nilpotency class at most n, then it is sometimes called a nil-n group. It follows immediately from any of the above forms of the definition of nilpotency, that the trivial group is the unique group of nilpotency class 0, and groups of nilpotency class 1 are exactly the non-trivial abelian groups. [2] [3]

  7. List of finite simple groups - Wikipedia

    en.wikipedia.org/wiki/List_of_finite_simple_groups

    F 4 (q) has a non-trivial graph automorphism when q is a power of 2. These groups are the automorphism groups of 8-dimensional Cayley algebras over finite fields, which gives them 7-dimensional representations. They also act on the corresponding Lie algebras of dimension 14. G 2 (q) has a non-trivial graph automorphism when q is a power of 3

  8. p-group - Wikipedia

    en.wikipedia.org/wiki/P-group

    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.

  9. Wikipedia : Naming conventions (chemistry)

    en.wikipedia.org/wiki/Wikipedia:Naming...

    The groups of the periodic table are numbered left to right as 1 – 18, with the f-block groups (columns) unnumbered, as recommended by IUPAC (1988). [1] Note that "group" has a chemical meaning and "column" has the tabular meaning. Alternatively, they may be named by their first element, e.g. "boron group" for group 13. [2]