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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.
English: The number on the rim of the node circle indicates an order of an element the node represents. Node's colour marks conjugacy class of element with exception that elements of group's center have the same colour (light gray) despite every one of them being of its own conjugacy class of one element.
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.
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 ...
For example, there are four non-isomorphic groups of order 16 that are semidirect products of C 8 and C 2; in this case, C 8 is necessarily a normal subgroup because it has index 2. One of these four semidirect products is the direct product, while the other three are non-abelian groups: the dihedral group of order 16; the quasidihedral group ...
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.
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 ...
Both these two groups and the dihedral group are semidirect products of a cyclic group <r> of order 2 n−1 with a cyclic group <s> of order 2. Such a non-abelian semidirect product is uniquely determined by an element of order 2 in the group of units of the ring / and there are precisely three such elements, , , and +, corresponding to the ...