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  2. 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

  3. List of small groups - Wikipedia

    en.wikipedia.org/wiki/List_of_small_groups

    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.

  4. Classification of finite simple groups - Wikipedia

    en.wikipedia.org/wiki/Classification_of_finite...

    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 ...

  5. 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.

  6. Free group - Wikipedia

    en.wikipedia.org/wiki/Free_group

    A free group of rank k clearly has subgroups of every rank less than k. Less obviously, a (nonabelian!) free group of rank at least 2 has subgroups of all countable ranks. The commutator subgroup of a free group of rank k > 1 has infinite rank; for example for F(a,b), it is freely generated by the commutators [a m, b n] for non-zero m and n ...

  7. 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 ...

  8. Database normalization - Wikipedia

    en.wikipedia.org/wiki/Database_normalization

    Every non-trivial functional dependency either begins with a superkey or ends with a prime attribute (attributes depend only on candidate keys) [5] Every non-trivial functional dependency either begins with a superkey or ends with an elementary prime attribute (a stricter form of 3NF) —

  9. Group by (SQL) - Wikipedia

    en.wikipedia.org/wiki/Group_by_(SQL)

    A GROUP BY statement in SQL specifies that a SQL SELECT statement partitions result rows into groups, based on their values in one or several columns. Typically, grouping is used to apply some sort of aggregate function for each group. [1] [2] The result of a query using a GROUP BY statement contains one row for