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  2. Abelian group - Wikipedia

    en.wikipedia.org/wiki/Abelian_group

    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.

  3. Normal subgroup - Wikipedia

    en.wikipedia.org/wiki/Normal_subgroup

    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 ...

  4. Glossary of group theory - Wikipedia

    en.wikipedia.org/wiki/Glossary_of_group_theory

    normal subgroup A subgroup N of a group G is normal in G (denoted N G) if the conjugation of an element n of N by an element g of G is always in N, that is, if for all g ∈ G and n ∈ N, gng −1 ∈ N. A normal subgroup N of a group G can be used to construct the quotient group G / N. normalizer

  5. Pure subgroup - Wikipedia

    en.wikipedia.org/wiki/Pure_subgroup

    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 ...

  6. Alternating group - Wikipedia

    en.wikipedia.org/wiki/Alternating_group

    A 5 is the smallest non-abelian simple group, having order 60, and thus the smallest non-solvable group. The group A 4 has the Klein four-group V as a proper normal subgroup, namely the identity and the double transpositions { (), (12)(34), (13)(24), (14)(23) }, that is the kernel of the surjection of A 4 onto A 3 ≅ Z 3.

  7. p-group - Wikipedia

    en.wikipedia.org/wiki/P-group

    Since every central subgroup is normal, it follows that every minimal normal subgroup of a finite p-group is central and has order p. Indeed, the socle of a finite p-group is the subgroup of the center consisting of the central elements of order p. If G is a p-group, then so is G/Z, and so it too has a non-trivial center.

  8. Torsion subgroup - Wikipedia

    en.wikipedia.org/wiki/Torsion_subgroup

    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.

  9. T-group (mathematics) - Wikipedia

    en.wikipedia.org/wiki/T-group_(mathematics)

    The solvable T-groups were characterized by Wolfgang Gaschütz as being exactly the solvable groups G with an abelian normal Hall subgroup H of odd order such that the quotient group G/H is a Dedekind group and H is acted upon by conjugation as a group of power automorphisms by G. A PT-group is a group in which permutability is transitive. A ...