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Every subgroup of an abelian group is normal, so each subgroup gives rise to a quotient group. Subgroups, quotients, and direct sums of abelian groups are again abelian. The finite simple abelian groups are exactly the cyclic groups of prime order. [6]: 32 The concepts of abelian group and -module agree.
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. The classification of nonabelian simple groups is far less trivial.
The solvable group 2 A 2 (2 2) is isomorphic to an extension of the order 8 quaternion group by an elementary abelian group of order 9. 2 A 2 (3 2) is isomorphic to the derived group G 2 (2)′. 2 A 3 (2 2) is isomorphic to B 2 (3). Remarks
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 ...
Order p 3: There are three abelian groups, and two non-abelian groups. One of the non-abelian groups is the semidirect product of a normal cyclic subgroup of order p 2 by a cyclic group of order p. The other is the quaternion group for p = 2 and a group of exponent p for p > 2.
This article gives a table of some common Lie groups and their associated Lie algebras.. The following are noted: the topological properties of the group (dimension; connectedness; compactness; the nature of the fundamental group; and whether or not they are simply connected) as well as on their algebraic properties (abelian; simple; semisimple).
Every elementary abelian p-group is a vector space over the prime field with p elements, and conversely every such vector space is an elementary abelian group. By the classification of finitely generated abelian groups, or by the fact that every vector space has a basis, every finite elementary abelian group must be of the form (Z/pZ) n for n a ...
If is such a group, and has order , then must be prime, since otherwise Cauchy's theorem applied to the (finite) subgroup generated by produces an element of order less than . Moreover, every finite subgroup of G {\displaystyle G} has order a power of p {\displaystyle p} (including G {\displaystyle G} itself, if it is finite).