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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 ...
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.
Any abelian group is metabelian. Any dihedral group is metabelian, as it has a cyclic normal subgroup of index 2. More generally, any generalized dihedral group is metabelian, as it has an abelian normal subgroup of index 2. If F is a field, the group of affine maps + (where a ≠ 0) acting on F is metabelian.
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 ...
Hall's universal group is a countable locally finite group containing each countable locally finite group as subgroup. Every group has a unique maximal normal locally finite subgroup (Robinson 1996, p. 436) Every periodic subgroup of the general linear group over the complex numbers is locally finite. Since all locally finite groups are ...
The (first) Ulm subgroup of an abelian group A, denoted U(A) or A 1, is p ω A = ∩ n p n A, where ω is the smallest infinite ordinal. It consists of all elements of A of infinite height. The family { U σ ( A )} of Ulm subgroups indexed by ordinals σ is defined by transfinite induction:
A minimal normal subgroup of a group G is a nontrivial normal subgroup N of G such that the only proper subgroup of N that is normal in G is the trivial subgroup. Every minimal normal subgroup of a group is characteristically simple. This follows from the fact that a characteristic subgroup of a normal subgroup is normal.