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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.
A normal subgroup of a normal subgroup of a group need not be normal in the group. That is, normality is not a transitive relation. The smallest group exhibiting this phenomenon is the dihedral group of order 8. [15] However, a characteristic subgroup of a normal subgroup is normal. [16] A group in which normality is transitive is called a T ...
Containment occurs exactly when S is abelian. If H is a subgroup of G, then N G (H) contains H. If H is a subgroup of G, then the largest subgroup of G in which H is normal is the subgroup N G (H). If S is a subset of G such that all elements of S commute with each other, then the largest subgroup of G whose center contains S is the subgroup C ...
In mathematics, specifically in group theory, an elementary abelian group is an abelian group in which all elements other than the identity have the same order. This common order must be a prime number , and the elementary abelian groups in which the common order is p are a particular kind of p -group .
Suppose that is a normal subgroup of a group , and / = {} denotes its set of cosets. Then there is a unique group law on G / N {\displaystyle G/N} for which the map G → G / N {\displaystyle G\to G/N} sending each element g {\displaystyle g} to g N {\displaystyle gN} is a homomorphism.
Similarly, a normal subgroup N of G is said to be a maximal normal subgroup (or maximal proper normal subgroup) of G if N < G and there is no normal subgroup K of G such that N < K < G. We have the following theorem: Theorem: A normal subgroup N of a group G is a maximal normal subgroup if and only if the quotient G/N is simple.
The torsion subgroup of an abelian group is pure. The directed union of pure subgroups is a pure subgroup. Since in a finitely generated abelian group the torsion subgroup is a direct summand, one might ask if the torsion subgroup is always a direct summand of an abelian group. It turns out that it is not always a summand, but it is a pure ...
An abelian group A is torsion-free if and only if it is flat as a Z-module, which means that whenever C is a subgroup of some abelian group B, then the natural map from the tensor product C ⊗ A to B ⊗ A is injective. Tensoring an abelian group A with Q (or any divisible group) kills torsion. That is, if T is a torsion group then T ⊗ Q = 0.