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  2. 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]

  3. Rank of a group - Wikipedia

    en.wikipedia.org/wiki/Rank_of_a_group

    If G is a finitely generated group, then the rank of G is a non-negative integer. The notion of rank of a group is a group-theoretic analog of the notion of dimension of a vector space. Indeed, for p-groups, the rank of the group P is the dimension of the vector space P/Φ(P), where Φ(P) is the Frattini subgroup.

  4. 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

  5. Extra special group - Wikipedia

    en.wikipedia.org/wiki/Extra_special_group

    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.

  6. Torsion-free abelian group - Wikipedia

    en.wikipedia.org/wiki/Torsion-free_abelian_group

    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 ...

  7. Rank of an abelian group - Wikipedia

    en.wikipedia.org/wiki/Rank_of_an_abelian_group

    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 ...

  8. Simple group - Wikipedia

    en.wikipedia.org/wiki/Simple_group

    Similarly, the additive group of the integers (, +) is not simple; the set of even integers is a non-trivial proper normal subgroup. [ 1 ] One may use the same kind of reasoning for any abelian group, to deduce that the only simple abelian groups are the cyclic groups of prime order.

  9. Simple Lie group - Wikipedia

    en.wikipedia.org/wiki/Simple_Lie_group

    Once these are known, the ones with non-trivial center are easy to list as follows. Any simple Lie group with trivial center has a universal cover whose center is the fundamental group of the simple Lie group. The corresponding simple Lie groups with non-trivial center can be obtained as quotients of this universal cover by a subgroup of the ...