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Order p 2: There are just two groups, both abelian. 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.
An abelian group is a set, together with an operation ･ , that combines any two elements and of to form another element of , denoted .The symbol ･ is a general placeholder for a concretely given operation.
The Cayley table tells us whether a group is abelian. Because the group operation of an abelian group is commutative, a group is abelian if and only if its Cayley table's values are symmetric along its diagonal axis. The group {1, −1} above and the cyclic group of order 3 under ordinary multiplication are both examples of abelian groups, and ...
Here, Z/pZ denotes the cyclic group of order p (or equivalently the integers mod p), and the superscript notation means the n-fold direct product of groups. [2] In general, a (possibly infinite) elementary abelian p-group is a direct sum of cyclic groups of order p. [4] (Note that in the finite case the direct product and direct sum coincide ...
Abelian varieties appear naturally as Jacobian varieties (the connected components of zero in Picard varieties) and Albanese varieties of other algebraic varieties. The group law of an abelian variety is necessarily commutative and the variety is non-singular. An elliptic curve is an abelian variety of dimension 1.
The Schur multiplier of the elementary abelian group of order 16 is an elementary abelian group of order 64, showing that the multiplier can be strictly larger than the group itself. The Schur multiplier of the quaternion group is trivial, but the Schur multiplier of dihedral 2-groups has order 2.
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.
Dedekind and Baer have shown (in the finite and respectively infinite order case) that every Hamiltonian group is a direct product of the form G = Q 8 × B × D, where B is an elementary abelian 2-group, and D is a torsion abelian group with all elements of odd order. Dedekind groups are named after Richard Dedekind, who investigated them in ...