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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.
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.
The additive group of a ring is an abelian group, but not all abelian groups are additive groups of rings (with nontrivial multiplication). Some important topics in this area of study are: Some important topics in this area of study are:
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 ...
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 ...
Pages in category "Abelian group theory" The following 37 pages are in this category, out of 37 total. This list may not reflect recent changes. 0–9. Abelian 2 ...