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In particular, any intermediate group Z n < A < Q n has rank n. 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 ...
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.
A non-finitely generated countable example is given by the additive group of the polynomial ring [] (the free abelian group of countable rank). More complicated examples are the additive group of the rational field Q {\displaystyle \mathbb {Q} } , or its subgroups such as Z [ p − 1 ] {\displaystyle \mathbb {Z} [p^{-1}]} (rational numbers ...
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 .
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.
Homology groups are finitely generated abelian groups, and homology classes are elements of these groups. The Betti numbers of the manifold are the rank of the free part of the homology group, and in the special case of surfaces, the torsion part of the homology group only occurs for non-orientable cycles.
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
A p-group G is called extraspecial if its center Z is cyclic of order p, and the quotient G/Z is a non-trivial elementary abelian p-group. Extraspecial groups of order p 1+2n are often denoted by the symbol p 1+2n. For example, 2 1+24 stands for an extraspecial group of order 2 25.