Search results
Results from the WOW.Com Content Network
The character theory of Z-groups is well understood (Çelik 1976), as they are monomial groups. The derived length of a Z-group is at most 2, so Z-groups may be insufficient for some uses. A generalization due to Hall are the A-groups, those groups with abelian Sylow subgroups.
List of all nonabelian groups up to order 31 Order Id. [a] G o i Group Non-trivial proper subgroups [1] Cycle graph Properties 6 7 G 6 1: D 6 = S 3 = Z 3 ⋊ Z 2: Z 3, Z 2 (3) : Dihedral group, Dih 3, the smallest non-abelian group, symmetric group, smallest Frobenius group.
Five of the eight group elements generate subgroups of order two, and the other two non-identity elements both generate the same cyclic subgroup of order four. In addition, there are two subgroups of the form Z 2 × Z 2, generated by pairs of order-two elements. The lattice formed by these ten subgroups is shown in the illustration.
Specifically, all subgroups of Z are of the form m = mZ, with m a positive integer. All of these subgroups are distinct from each other, and apart from the trivial group {0} = 0Z, they all are isomorphic to Z. The lattice of subgroups of Z is isomorphic to the dual of the lattice of natural numbers ordered by divisibility. [10]
There is a natural homomorphism SL(2, Z) → SL(2, Z/NZ) given by reducing the entries modulo N. This induces a homomorphism on the modular group PSL(2, Z) → PSL(2, Z/NZ). The kernel of this homomorphism is called the principal congruence subgroup of level N, denoted Γ(N). We have the following short exact sequence:
Dih n = Dih(Z n) (the dihedral groups) . For even n there are two sets {(h + k + k, 1) | k in H}, and each generates a normal subgroup of type Dih n / 2.As subgroups of the isometry group of the set of vertices of a regular n-gon they are different: the reflections in one subgroup all have two fixed points, while none in the other subgroup has (the rotations of both are the same).
The union of subgroups A and B is a subgroup if and only if A ⊆ B or B ⊆ A. A non-example: 2 Z ∪ 3 Z {\displaystyle 2\mathbb {Z} \cup 3\mathbb {Z} } is not a subgroup of Z , {\displaystyle \mathbb {Z} ,} because 2 and 3 are elements of this subset whose sum, 5, is not in the subset.
In the case of n = 2 this gives the rather obvious result that a subgroup H of index 2 is a normal subgroup, because the normal subgroup of H must have index 2 in G and therefore be identical to H. (We can arrive at this fact also by noting that all the elements of G that are not in H constitute the right coset of H and also the left coset, so ...